[{"data":1,"prerenderedAt":16935},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":374,"course-wordcounts":16803,"ref-card-index":16859},[4,28,50,71,120,152,205,230,286,306,331,352],{"module":5,"moduleNumber":6,"slug":7,"lessons":8},"Foundations",1,"foundations",[9,15,21],{"title":10,"path":11,"lessonNumber":6,"topics":12,"summary":14},"What Is an Algorithm?","\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm",[5,13],"Correctness & Induction","An algorithm is a finite, mechanical recipe that transforms inputs into outputs. We pin down what counts as an algorithm, how we write one down, and the three things we always ask of it: is it correct, is it fast, and can we prove it.\n",{"title":16,"path":17,"lessonNumber":18,"topics":19,"summary":20},"Asymptotic Analysis","\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis",2,[16],"We measure an algorithm's running time as a function of its input size, then strip away machine-specific constants and lower-order terms to compare algorithms cleanly. This lesson defines the RAM model and the $O$, $\\Omega$, $\\Theta$, $o$, and $\\omega$ notations, and shows how to read the cost of loops off the page.\n",{"title":22,"path":23,"lessonNumber":24,"topics":25,"summary":27},"Recurrences and the Master Theorem","\u002Falgorithms\u002Ffoundations\u002Frecurrences",3,[26],"Recurrences","Recursive and divide-and-conquer algorithms describe their own running time with a recurrence: $T(n)$ in terms of $T$ on smaller inputs. We solve recurrences three ways — drawing the recursion tree, guessing-and-verifying by induction, and applying the Master Theorem — using merge sort as the running example.\n",{"module":29,"moduleNumber":18,"slug":30,"lessons":31},"Divide & Conquer","divide-and-conquer",[32,38,44],{"title":33,"path":34,"lessonNumber":6,"topics":35,"summary":37},"Divide and Conquer & Mergesort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort",[29,36],"Comparison Sorting","Divide and conquer breaks a problem into smaller copies of itself, solves them recursively, and stitches the answers together. We meet the paradigm through mergesort — its merge step, its loop-invariant proof, and the recursion tree that pins its cost at $\\Theta(n\\log n)$ — then glimpse Karatsuba multiplication as a second example of the same idea.",{"title":39,"path":40,"lessonNumber":18,"topics":41,"summary":43},"Quicksort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort",[36,42],"Probabilistic Analysis","Quicksort sorts in place by partitioning around a pivot and recursing on each side. We give Lomuto and Hoare partitioning with a correctness invariant, see why a bad pivot costs $\\Theta(n^2)$ while a balanced one gives $\\Theta(n\\log n)$, and prove that randomizing the pivot makes the expected cost $\\Theta(n\\log n)$ on every input.",{"title":45,"path":46,"lessonNumber":24,"topics":47,"summary":49},"Linear-Time Selection","\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection",[48,29],"Order Statistics","Finding the $k$-th smallest element looks like it should require sorting, but it does not. Quickselect adapts quicksort's partition to recurse on just one side, achieving expected $O(n)$. The median-of-medians algorithm guarantees a good pivot with the groups-of-five trick, pushing the worst case down to a provable $O(n)$.",{"module":51,"moduleNumber":24,"slug":52,"lessons":53},"Sorting & Order Statistics","sorting",[54,60,65],{"title":55,"path":56,"lessonNumber":6,"topics":57,"summary":59},"Heaps and Heapsort","\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort",[58,36],"Heaps","A binary heap is a tree we store flat in an array, with index arithmetic standing in for pointers. We build the max-heap property bottom-up in $O(n)$ time, sort in place in $\\Theta(n\\log n)$ by repeatedly extracting the maximum, and reuse the same structure to implement a priority queue.",{"title":61,"path":62,"lessonNumber":18,"topics":63,"summary":64},"Lower Bounds for Comparison Sorting","\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds",[36],"Every sort we have seen runs in $\\Omega(n\\log n)$, and that is no accident. Modeling a sort as a decision tree of comparisons, we show any such tree must have $n!$ leaves, forcing height $\\ge \\log_2(n!) = \\Omega(n\\log n)$ — a worst-case bound no comparison sort can ever beat.",{"title":66,"path":67,"lessonNumber":24,"topics":68,"summary":70},"Sorting in Linear Time","\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting",[69],"Linear-Time Sorting","The $\\Omega(n\\log n)$ barrier only binds algorithms that compare. By instead using keys as array indices we slip past it: counting sort runs in $\\Theta(n+k)$ and is stable, radix sort layers it digit by digit, and bucket sort averages $\\Theta(n)$ on uniform data. We see exactly when each applies.",{"module":72,"moduleNumber":73,"slug":74,"lessons":75},"Data Structures",4,"data-structures",[76,82,88,93,99,105,113],{"title":77,"path":78,"lessonNumber":6,"topics":79,"summary":81},"Elementary Data Structures","\u002Falgorithms\u002Fdata-structures\u002Felementary-structures",[80],"Linear Structures","Every container is built one of two ways: **contiguous** in an array, or **linked** through pointers. We trade cache-friendly random access against $O(1)$ splicing, derive the **amortized $O(1)$** append of a doubling dynamic array, and assemble the two ordered access disciplines — the LIFO **stack** and the FIFO **queue** (with its generalization, the **deque**) — on top of both.",{"title":83,"path":84,"lessonNumber":18,"topics":85,"summary":87},"Hash Tables","\u002Falgorithms\u002Fdata-structures\u002Fhash-tables",[86],"Hashing","A hash table implements the dictionary — insert, search, delete — in expected $O(1)$ time by scattering keys across an array with a hash function. We build up from direct addressing, handle collisions by chaining and by open addressing, analyze the load factor $\\alpha$, and see how universal hashing earns its expected-time guarantee against every input.",{"title":89,"path":90,"lessonNumber":24,"topics":91,"summary":92},"Binary Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees",[89],"A binary search tree keeps keys ordered so that every operation follows a single root-to-leaf path. We state the BST property, give search and insert, find minimum, maximum, and successor, see that an inorder walk emits the keys in sorted order, and confront the catch — every operation costs $O(h)$, and a carelessly built tree degrades to height $h = \\Theta(n)$, motivating balance.",{"title":94,"path":95,"lessonNumber":73,"topics":96,"summary":98},"AVL Trees","\u002Falgorithms\u002Fdata-structures\u002Favl-trees",[97],"Balanced Trees","An AVL tree is the first balanced BST: at every node the two subtrees' heights differ by at most $1$. A Fibonacci-style minimal-node argument forces height $h \\le 1.44\\log_2 n = O(\\log n)$, so search, insert, and delete are all $O(\\log n)$. Insertion rebalances with at most one of four rotation cases (LL, RR, LR, RL); deletion may rotate all the way to the root.",{"title":100,"path":101,"lessonNumber":102,"topics":103,"summary":104},"Balanced Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees",5,[97],"An ordinary BST can degrade to height $\\Theta(n)$; balanced search trees guarantee $h = O(\\log n)$ by maintaining invariants and repairing them after every update. We meet rotations, the local restructuring primitive, then red-black trees, whose color invariants force logarithmic height, and finally B-trees, which trade tall-and-thin for short-and-wide to win on disk.",{"title":106,"path":107,"lessonNumber":108,"topics":109,"summary":112},"Disjoint Sets (Union-Find)","\u002Falgorithms\u002Fdata-structures\u002Funion-find",6,[110,111],"Disjoint Sets","Amortized Analysis","The disjoint-set data structure tracks a partition of elements into groups, answering \"are these two in the same group?\" and merging groups on demand. A forest of parent pointers, sped up by union by rank and path compression, drives every operation to near-constant $O(\\alpha(n))$ amortized time — the engine behind connectivity queries and Kruskal's minimum spanning tree.",{"title":114,"path":115,"lessonNumber":116,"topics":117,"summary":119},"Fenwick & Segment Trees","\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees",7,[118],"Range Queries","A prefix-sum array answers a range sum in $O(1)$ but pays $O(n)$ per update; a plain array updates in $O(1)$ but pays $O(n)$ per range sum. Fenwick and segment trees give us _both_ in $O(\\log n)$. The Fenwick (binary indexed) tree is a tiny array keyed by the low bit; the segment tree is a general balanced tree over canonical ranges that handles any associative aggregate and, with lazy propagation, range updates too.",{"module":121,"moduleNumber":102,"slug":122,"lessons":123},"Sequences & Strings","sequences",[124,130,135,141,147],{"title":125,"path":126,"lessonNumber":6,"topics":127,"summary":129},"Two Pointers, Sliding Windows & Prefix Sums","\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows",[128],"Array Techniques","A family of array idioms that collapse an obvious $O(n^2)$ scan into a single $O(n)$ pass by maintaining an invariant as indices move. We meet two pointers (converging on a sorted array, and a fast\u002Fslow pair for in-place rewriting), the sliding window (fixed and variable size, amortized $O(n)$), and prefix sums, which answer any range-sum in $O(1)$ and count subarrays summing to $k$.",{"title":131,"path":132,"lessonNumber":18,"topics":133,"summary":134},"Monotonic Stacks & Queues","\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks",[128],"A **monotonic stack** keeps its contents sorted by popping every element that would break the order before each push — turning a family of \"previous\u002Fnext greater (or smaller) element\" questions into a single $O(n)$ scan. We derive the next-greater-element routine and its amortized analysis, fuse two such scans to measure the **largest rectangle in a histogram** in linear time, and extend the idea to a **monotonic deque** that streams the **sliding-window maximum** in $O(n)$.",{"title":136,"path":137,"lessonNumber":24,"topics":138,"summary":140},"Binary Search on the Answer","\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer",[139],"Searching","Binary search is not really about arrays — it is about locating the boundary of a **monotone predicate** $p(x)$ in $O(\\log(\\text{range}))$ probes. We first pin down the half-open `while (lo \u003C hi)` template for $\\textsc{lower\\_bound}$ and $\\textsc{upper\\_bound}$, then generalize to \"binary search on the answer\": whenever feasibility is monotone in a numeric parameter, we binary search the parameter itself, calling a feasibility check at each step.",{"title":142,"path":143,"lessonNumber":73,"topics":144,"summary":146},"String Matching: Rabin–Karp, KMP & Z","\u002Falgorithms\u002Fsequences\u002Fstring-matching",[145],"Strings","Given a text $T$ of length $n$ and a pattern $P$ of length $m$, find every occurrence of $P$ in $T$. The naive scan costs $O(nm)$; we beat it three ways. Rabin–Karp uses a **rolling hash** to test alignments in $O(1)$ amortised each, with expected $O(n+m)$. KMP precomputes a **failure function** so the scan never re-reads a text character, for worst-case $O(n+m)$. The **Z-function** gives the same bound from a different angle and converts freely to KMP's table.",{"title":148,"path":149,"lessonNumber":102,"topics":150,"summary":151},"Tries & Prefix Trees","\u002Falgorithms\u002Fsequences\u002Ftries",[145],"A **trie** stores a set of strings in a tree keyed by _characters_, so that insert, search, and prefix-test all run in $O(L)$ time — the length of the key, _independent of how many keys are stored_. Shared prefixes are stored once, which makes tries the natural structure for autocomplete, wildcard dictionaries, board word-search, and — over the alphabet $\\{0,1\\}$ — the maximum-XOR-pair problem.",{"module":153,"moduleNumber":108,"slug":154,"lessons":155},"Graphs","graphs",[156,163,168,173,178,183,188,193,199],{"title":157,"path":158,"lessonNumber":6,"topics":159,"summary":162},"Graph Representations and Traversal","\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal",[160,161],"Graph Representations","Graph Traversal","A graph captures _relationships_ — who connects to whom. We fix the vocabulary, weigh the two standard representations (adjacency list versus matrix), then meet the two explorations you'll use constantly: breadth-first search, which finds shortest paths by number of edges, and depth-first search, whose discovery and finish times reveal a graph's hidden structure. Both run in $O(V + E)$.",{"title":164,"path":165,"lessonNumber":18,"topics":166,"summary":167},"Topological Sort and Strong Connectivity","\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc",[161],"Directed acyclic graphs model dependencies: tasks that must precede other tasks. A _topological order_ lays such a graph out in a line so every edge points forward, and depth-first finish times hand it to us almost for free. We then ask the harder question for graphs _with_ cycles: which vertices can reach each other? The answer is the strongly connected components, found by a two-pass DFS.",{"title":169,"path":170,"lessonNumber":24,"topics":171,"summary":172},"Minimum Spanning Trees","\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees",[169],"Given a weighted network, how do we connect everything as cheaply as possible? The answer is a minimum spanning tree. One lemma, the cut property, justifies _every_ correct MST algorithm, and from it two famous greedy methods fall out: Kruskal's, which grows a forest edge by edge with a union-find structure, and Prim's, which grows a single tree using a priority queue.",{"title":174,"path":175,"lessonNumber":73,"topics":176,"summary":177},"Shortest Paths","\u002Falgorithms\u002Fgraphs\u002Fshortest-paths",[174],"Finding the cheapest route through a weighted network is one of the most-used algorithms in computing. A single operation — _relaxation_ — underlies them all. Dijkstra's algorithm solves the non-negative case greedily; Bellman-Ford handles negative edges and detects negative cycles; and Floyd-Warshall finds the shortest path between _every_ pair of vertices.",{"title":179,"path":180,"lessonNumber":102,"topics":181,"summary":182},"Network Flow","\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow",[179],"How much can flow through a network from source to sink? Max-flow is a surprisingly general model — once you see a problem as flow, a whole toolbox opens up. We build flow networks, find maximum flows by repeatedly pushing along augmenting paths in the residual graph, prove the max-flow min-cut theorem, and watch bipartite matching fall out as a special case.",{"title":184,"path":185,"lessonNumber":108,"topics":186,"summary":187},"Bridges & Articulation Points","\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points",[153],"A **bridge** is an edge whose removal disconnects the graph; an **articulation point** is a vertex whose removal does. Both are single points of failure in a network. A single depth-first search computes discovery times and **low-links**, and two local criteria — $low[v] > disc[u]$ for bridges, $low[v] \\ge disc[u]$ for cut vertices — find them all in $O(V+E)$.",{"title":189,"path":190,"lessonNumber":116,"topics":191,"summary":192},"Lowest Common Ancestor & Binary Lifting","\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor",[153],"Given a rooted tree, the lowest common ancestor of $u$ and $v$ is the deepest node that is an ancestor of both. A naive walk answers one query in $O(h)$; **binary lifting** precomputes the $2^k$-th ancestor of every node in $O(n\\log n)$, then answers $k$-th-ancestor and LCA queries in $O(\\log n)$ each. We derive both jumps, apply them to tree distance, and compare against the Euler-tour + RMQ and Tarjan offline alternatives.",{"title":194,"path":195,"lessonNumber":196,"topics":197,"summary":198},"2-SAT via Implication Graphs","\u002Falgorithms\u002Fgraphs\u002Ftwo-sat",8,[153],"A boolean formula whose every clause has exactly two literals can be solved in _linear_ time — even though its three-literal cousin is NP-complete. The trick is to read each clause as a pair of implications, build a directed graph on the $2n$ literals, and ask a question we already know how to answer: which literals share a strongly connected component? The formula is satisfiable iff no variable lands in the same SCC as its own negation, and the SCCs' topological order hands us a satisfying assignment for free.",{"title":200,"path":201,"lessonNumber":202,"topics":203,"summary":204},"Eulerian Tours","\u002Falgorithms\u002Fgraphs\u002Feulerian-tours",9,[153],"An **Eulerian tour** uses every _edge_ of a graph exactly once. We give the exact parity and balance conditions under which one exists (even degree for undirected graphs, in-degree equal to out-degree for directed) and Hierholzer's $O(E)$ algorithm that constructs one by splicing closed sub-tours. We contrast this sharply with the **Hamiltonian** problem (visit every _vertex_ once), which is NP-complete: visiting edges is easy, visiting vertices is hard.",{"module":206,"moduleNumber":116,"slug":207,"lessons":208},"Greedy Algorithms","greedy",[209,214,220,225],{"title":210,"path":211,"lessonNumber":6,"topics":212,"summary":213},"The Greedy Method","\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method",[206],"A greedy algorithm builds a solution one locally-best choice at a time and never looks back. We pin down the two properties that make this work — the greedy-choice property and optimal substructure — prove the canonical activity-selection algorithm correct with an exchange argument, watch greedy fail spectacularly on the 0\u002F1 knapsack, and glimpse matroids as the theory that says exactly when greed is good.",{"title":215,"path":216,"lessonNumber":18,"topics":217,"summary":219},"Scheduling & Interval Partitioning","\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals",[218],"Greedy","Three classic scheduling problems all yield to greedy algorithms — and all three turn on a single design decision: which key to sort by. Interval scheduling sorts by **finish** time to pack the most compatible jobs; interval partitioning sorts by **start** time and proves the rooms needed equal the maximum overlap **depth**; minimizing maximum lateness sorts by **deadline** and is justified by an adjacent-swap exchange argument.",{"title":221,"path":222,"lessonNumber":24,"topics":223,"summary":224},"Huffman Codes","\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes",[206],"Huffman coding is the greedy method's most beautiful application: it builds a provably optimal prefix-free binary code by repeatedly merging the two least frequent symbols. We develop prefix-free codes as binary trees, give the algorithm with a priority queue, build a Huffman tree from example frequencies, prove optimality with the same greedy-choice-plus-substructure argument, and pin the running time at $O(n\\log n)$.",{"title":226,"path":227,"lessonNumber":73,"topics":228,"summary":229},"Matroids & Exchange Arguments","\u002Falgorithms\u002Fgreedy\u002Fmatroids",[218],"The capstone of the greedy module: _why_ and _when_ a greedy algorithm is provably optimal. We recap the two correctness templates — **greedy-stays-ahead** and the **exchange argument** — then meet the **matroid** $M=(S,\\mathcal{I})$, an abstraction whose **exchange property** is exactly the structure greedy needs. The matroid–greedy theorem says sorting by weight and taking what stays independent yields a maximum-weight basis _if and only if_ the structure is a matroid. Kruskal's MST is the canonical instance; 0\u002F1 knapsack and TSP are the canonical failures.",{"module":231,"moduleNumber":196,"slug":232,"lessons":233},"Dynamic Programming","dynamic-programming",[234,239,245,250,255,260,265,270,275,280],{"title":235,"path":236,"lessonNumber":6,"topics":237,"summary":238},"Principles of Dynamic Programming","\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples",[231,26],"Dynamic programming is recursion with memory: when a recursive solution re-solves the same subproblems again and again, we solve each one once and store the answer. We pin down the two structural conditions that make this work — overlapping subproblems and optimal substructure — contrast top-down memoization with bottom-up tabulation, and distil the whole method into a five-step recipe.",{"title":240,"path":241,"lessonNumber":18,"topics":242,"summary":244},"Sequence Alignment & LCS","\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp",[231,243],"String Structures","Two strings can be compared by asking how much of one survives inside the other. The longest common subsequence (LCS) and edit distance are the two classic answers, and they are the _same_ dynamic program wearing different costs. We derive the LCS recurrence by examining the last characters, fill a worked DP table, reconstruct the subsequence, and then show edit distance as the identical $\\Theta(mn)$ pattern.",{"title":246,"path":247,"lessonNumber":24,"topics":248,"summary":249},"Longest Increasing Subsequence","\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence",[231],"Given a sequence of numbers, how long is its longest strictly increasing subsequence? A first dynamic program indexes subproblems by the element each subsequence _ends at_, giving an $O(n^2)$ solution with parent-pointer reconstruction. A sharper idea, the patience-sorting _tails_ array searched by binary search, drops the time to $O(n\\log n)$. We then fold in the variants: non-decreasing, counting, Russian-doll envelopes, and bitonic.",{"title":251,"path":252,"lessonNumber":73,"topics":253,"summary":254},"Knapsack & Subset Problems","\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack",[231],"We start from $\\textsc{Subset-sum}$ — does some sublist hit a target $t$? — and its include\u002Fexclude recurrence over a boolean table $A(i, u)$, then bolt on values to get 0\u002F1 knapsack as the same machine with $\\lor$ promoted to $\\max$. We fill both tables, recover the chosen items, and confront the surprise that the $\\Theta(nt)$ running time is only _pseudo-polynomial_ — exponential in the bit length $b$, and unimprovable unless $\\mathrm{P}=\\mathrm{NP}$ since subset-sum is $\\textsc{NP-complete}$. The fractional variant reveals the sharp line between greedy and dynamic programming.",{"title":256,"path":257,"lessonNumber":102,"topics":258,"summary":259},"Coin Change & Unbounded Knapsack","\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded",[231],"The previous lesson let each item be taken at most once. Drop that cap — items may be reused _any number of times_ — and the 0\u002F1 knapsack collapses from a two-dimensional table to a one-dimensional one, because there is no longer a prefix of \"already-used\" items to track. We meet **unbounded knapsack**, then its most famous instance, **coin change**: the minimum-coins recurrence $C[a] = 1 + \\min_c C[a-c]$, and the counting variant where the _order of the loops_ decides whether you count unordered combinations or ordered sequences — the classic bug. Greed fails in general but works for canonical coin systems.",{"title":261,"path":262,"lessonNumber":108,"topics":263,"summary":264},"Interval DP","\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp",[231],"Many problems ask for the best way to combine a contiguous range of items, and the answer is a dynamic program over subintervals $[i,j]$ that chooses a split point $k$. We derive the pattern from matrix-chain multiplication — parenthesising a product to minimize scalar multiplications in $O(n^3)$ — distil it into a reusable template filled by increasing interval length, and then meet its sharpest variant: the \"last operation\" trick behind Burst Balloons and cutting a stick, where fixing the _last_ move (not the first) makes the two sides independent.",{"title":266,"path":267,"lessonNumber":116,"topics":268,"summary":269},"Dynamic Programming on Trees","\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp",[231],"When the subproblems of a dynamic program are _rooted subtrees_, a single post-order DFS solves the whole thing in $O(n)$: each node combines the already-computed answers of its children. We meet the archetype — maximum-weight independent set on a tree — then the \"path through a node\" pattern behind tree diameter and maximum path sum, and finally **rerooting**, which computes a per-node answer for _every_ node as root in $O(n)$ with two passes.",{"title":271,"path":272,"lessonNumber":196,"topics":273,"summary":274},"Bitmask DP","\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp",[231],"When a subproblem depends not on an index or a prefix but on _which subset_ of a small ground set has been used, we can encode that subset as the bits of an integer and index a DP table by it. With $n \\le \\sim 20$ the $2^n$ subsets fit in a table, turning $\\Theta(n!)$ brute force into $O(2^n \\cdot \\text{poly}(n))$. We meet the bit tricks, the Held–Karp TSP archetype, assignment by mask, subset-sum partitioning, and submask enumeration with its $3^n$ bound.",{"title":276,"path":277,"lessonNumber":202,"topics":278,"summary":279},"DP Optimizations","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations",[231],"A correct DP recurrence is only half the battle; its naive evaluation is often a factor of $n$ slower than necessary. This capstone surveys five techniques, monotonic-queue, the convex hull trick, divide-and-conquer optimization, Knuth's optimization, and SOS DP, that each exploit _structure in the transition_ (a sliding window, linear costs, monotone optimal splits, the quadrangle inequality, or subset lattices) to shave an $O(n)$, $O(\\log n)$, or worse factor off the running time.",{"title":281,"path":282,"lessonNumber":283,"topics":284,"summary":285},"Dynamic Programming on Graphs","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs",10,[231],"Many graph algorithms are dynamic programs in disguise: the subproblem is the _best value reachable under a restricted resource_ — intermediate vertices allowed, edges allowed, or a topological prefix — and edge _relaxation_ is the DP transition. We frame Floyd–Warshall as the archetype ($O(V^3)$ all-pairs shortest paths), Bellman–Ford as a DP over path length (the at-most-$K$-stops variant), DAG-DP in topological order ($O(V+E)$), and Warshall's transitive closure as the boolean analog.",{"module":287,"moduleNumber":202,"slug":288,"lessons":289},"Backtracking & Search","backtracking",[290,296,301],{"title":291,"path":292,"lessonNumber":6,"topics":293,"summary":295},"Backtracking: Subsets, Permutations & Combinations","\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals",[294],"Backtracking","Backtracking builds a solution one choice at a time and abandons a partial solution the moment it cannot be completed, exploring a state-space tree by depth-first search. We meet the universal choose\u002Fexplore\u002Fun-choose template, derive the canonical enumerations — subsets ($2^n$), permutations ($n!$), and combinations ($\\binom{n}{k}$) — handle duplicate elements by skipping equal siblings, and see how pruning turns an exponential search into a tractable one.",{"title":297,"path":298,"lessonNumber":18,"topics":299,"summary":300},"Constraint Search: N-Queens & Sudoku","\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search",[294],"Many hard puzzles are **constraint satisfaction problems**: assign each variable a value from its domain so that every constraint holds. Backtracking solves them by assigning variables one at a time and rejecting a partial assignment the instant a constraint breaks. We make the rejection cheap — $O(1)$ conflict checks for N-Queens via column and diagonal sets — and prune harder with **forward checking**, **MRV** ordering, and **constraint propagation**, which is what lets an exponential search actually finish.",{"title":302,"path":303,"lessonNumber":24,"topics":304,"summary":305},"Branch & Bound and Meet in the Middle","\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound",[294],"Plain backtracking prunes a search tree by _feasibility_; for _optimization_ problems we can prune far more aggressively by _value_. **Branch and bound** keeps the best complete solution found so far and discards any partial solution whose optimistic bound cannot beat it. **Meet in the middle** splits the instance in two, enumerates each half, and recombines by binary search — turning $2^n$ into $O(2^{n\u002F2}\\,n)$ and pushing exact search out to $n \\approx 40$.",{"module":307,"moduleNumber":283,"slug":308,"lessons":309},"Mathematical Algorithms","mathematical-algorithms",[310,316,321,326],{"title":311,"path":312,"lessonNumber":6,"topics":313,"summary":315},"Number Theory: GCD & Modular Arithmetic","\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics",[314],"Number Theory","This lesson opens the mathematical-algorithms module with the bedrock of computational number theory. We prove Euclid's recurrence $\\gcd(a,b)=\\gcd(b,\\,a\\bmod b)$ and its $O(\\log\\min(a,b))$ running time, extend it to recover Bézout coefficients $x,y$ with $ax+by=\\gcd(a,b)$, and build modular arithmetic on residue classes — including when a modular inverse $a^{-1}\\bmod m$ exists and how to compute it.",{"title":317,"path":318,"lessonNumber":18,"topics":319,"summary":320},"Modular Exponentiation & Primality","\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality",[314],"Computing $a^n \\bmod m$ naively costs $n$ multiplications; **repeated squaring** does it in $O(\\log n)$ by reading the bits of the exponent. We use this routine to state **Fermat's little theorem** (and the modular inverse it gives), then to test primality — trial division, the probabilistic **Fermat** and **Miller–Rabin** tests, and the deterministic witness set that settles primality for every 64-bit number.",{"title":322,"path":323,"lessonNumber":24,"topics":324,"summary":325},"Sieves & Factorization","\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization",[314],"The previous lesson tested one number for primality; here we ask for _all_ primes up to $n$ at once. The **sieve of Eratosthenes** cross-cuts composites in $O(n\\log\\log n)$, and a **linear sieve** does it in $O(n)$ while recording each number's **smallest prime factor**, which then factors any $x \\le n$ in $O(\\log x)$. From a factorization $x = \\prod p_i^{e_i}$ the multiplicative functions $\\tau$, $\\sigma$, and Euler's totient $\\varphi$ fall out immediately.",{"title":327,"path":328,"lessonNumber":73,"topics":329,"summary":330},"Combinatorics & Counting","\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics",[314],"Counting is the arithmetic of finite sets. We build up from permutations $n!$ and combinations $\\binom{n}{k}$, prove Pascal's rule by a bijection, and count multisets with stars and bars. The practical core is computing $\\binom{n}{k}\\bmod p$ in $O(1)$ from precomputed factorials and inverse factorials. We close with inclusion–exclusion and the Chinese Remainder Theorem, both of which lean on the modular inverse from the previous lesson.",{"module":332,"moduleNumber":333,"slug":334,"lessons":335},"Computational Geometry",11,"computational-geometry",[336,342,347],{"title":337,"path":338,"lessonNumber":6,"topics":339,"summary":341},"Geometric Primitives & Orientation","\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives",[340],"Geometry","Computational geometry is built on a single reliable primitive — the **orientation test**, a sign of a cross product that tells whether three points turn left, right, or lie collinear. From points-as-vectors and the dot and cross products we derive orientation, segment intersection, the shoelace area formula, and point-in-polygon tests, keeping all arithmetic **exact and integer** so that no floating-point rounding can corrupt a sign.",{"title":343,"path":344,"lessonNumber":18,"topics":345,"summary":346},"Convex Hull","\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull",[340],"The convex hull is the smallest convex polygon enclosing a point set — the rubber band snapped around the nails. We build it with Andrew's monotone chain, sorting by $(x,y)$ and sweeping a lower and upper hull while popping any non-left turn via the orientation primitive, in $O(n\\log n)$. A reduction from sorting shows that bound is optimal, and the hull unlocks diameter, smallest enclosing rectangle, and more through rotating calipers.",{"title":348,"path":349,"lessonNumber":24,"topics":350,"summary":351},"Sweep-Line Algorithms","\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line",[340],"The plane-sweep paradigm turns a static $2$-D geometry problem into a dynamic $1$-D ordered-set problem: a vertical line sweeps left to right, stopping at an $x$-sorted **event queue** while a balanced-BST **status structure** tracks the objects it currently crosses, ordered by $y$. We derive Bentley–Ottmann segment intersection in $O((n+k)\\log n)$, recover closest-pair in $O(n\\log n)$, and reduce skyline, rectangle-area, and overlap problems to $\\pm1$ event sweeps.",{"module":353,"moduleNumber":354,"slug":355,"lessons":356},"Intractability",12,"intractability",[357,363,367],{"title":358,"path":359,"lessonNumber":6,"topics":360,"summary":362},"P, NP, and Reductions","\u002Falgorithms\u002Fintractability\u002Fp-np-reductions",[361],"NP-Completeness","Most problems we have met so far have fast algorithms. A vast and important family seemingly does not. This lesson builds the vocabulary for that divide: decision problems, the class $\\mathsf{P}$ of problems we can solve quickly, the class $\\mathsf{NP}$ of problems whose solutions we can _check_ quickly, and polynomial-time reductions, the tool that lets us compare the difficulty of two problems without solving either.",{"title":361,"path":364,"lessonNumber":18,"topics":365,"summary":366},"\u002Falgorithms\u002Fintractability\u002Fnp-completeness",[361],"Some problems in $\\mathsf{NP}$ are universally hardest: every other problem in $\\mathsf{NP}$ reduces to them. This lesson defines $\\mathsf{NP}$-hard and $\\mathsf{NP}$-complete, states the Cook–Levin theorem that anchors the whole edifice on **SAT**, walks the web of reductions that grows from it, and gives the four-step recipe for proving a brand-new problem $\\mathsf{NP}$-complete.",{"title":368,"path":369,"lessonNumber":24,"topics":370,"summary":373},"Coping with NP-Hardness","\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness",[371,372],"Approximation","Heuristics","Proving a problem $\\mathsf{NP}$-hard is the beginning, not the end. The world still needs answers. This lesson surveys the four honest responses to hardness: approximation algorithms with a provable ratio (worked through a 2-approximation for vertex cover), heuristics and local search, exact exponential methods like branch and bound, and exploiting special structure in the instances you actually face.",{"id":375,"title":327,"blurb":376,"body":377,"description":16766,"extension":16767,"meta":16768,"module":307,"navigation":16770,"path":328,"practice":16771,"rawbody":16784,"readingTime":16785,"seo":16790,"sources":16791,"status":16799,"stem":16800,"summary":330,"topics":16801,"__hash__":16802},"course\u002F01.algorithms\u002F10.mathematical-algorithms\u002F04.combinatorics.md","",{"type":378,"value":379,"toc":16753},"minimark",[380,577,582,644,803,856,1100,1203,1482,2013,2018,2021,2352,2809,2816,3266,3769,3903,4215,4219,4222,4535,4921,4925,5276,5391,5609,5908,6213,6468,6957,7070,7446,7691,7736,7984,8271,8393,9036,9040,9047,10063,10306,10470,11030,11251,11692,11696,11706,12139,13164,13448,14230,14845,14890,14909,14913,16225,16749],[381,382,383,387,388,392,393,564,565,568,569,572,573,576],"p",{},[384,385,386],"a",{"href":318},"Modular exponentiation"," and, through Fermat's little\ntheorem, the ",[389,390,391],"strong",{},"modular inverse"," ",[394,395,398],"span",{"className":396},[397],"katex",[394,399,403,475,529],{"className":400,"ariaHidden":402},[401],"katex-html","true",[394,404,407,412,462,467,472],{"className":405},[406],"base",[394,408],{"className":409,"style":411},[410],"strut","height:0.8141em;",[394,413,416,420],{"className":414},[415],"mord",[394,417,384],{"className":418},[415,419],"mathnormal",[394,421,424],{"className":422},[423],"msupsub",[394,425,428],{"className":426},[427],"vlist-t",[394,429,432],{"className":430},[431],"vlist-r",[394,433,436],{"className":434,"style":411},[435],"vlist",[394,437,439,444],{"style":438},"top:-3.063em;margin-right:0.05em;",[394,440],{"className":441,"style":443},[442],"pstrut","height:2.7em;",[394,445,451],{"className":446},[447,448,449,450],"sizing","reset-size6","size3","mtight",[394,452,454,458],{"className":453},[415,450],[394,455,457],{"className":456},[415,450],"−",[394,459,461],{"className":460},[415,450],"1",[394,463],{"className":464,"style":466},[465],"mspace","margin-right:0.2778em;",[394,468,471],{"className":469},[470],"mrel","≡",[394,473],{"className":474,"style":466},[465],[394,476,478,481,521,525],{"className":477},[406],[394,479],{"className":480,"style":411},[410],[394,482,484,487],{"className":483},[415],[394,485,384],{"className":486},[415,419],[394,488,490],{"className":489},[423],[394,491,493],{"className":492},[427],[394,494,496],{"className":495},[431],[394,497,499],{"className":498,"style":411},[435],[394,500,501,504],{"style":438},[394,502],{"className":503,"style":443},[442],[394,505,507],{"className":506},[447,448,449,450],[394,508,510,513,517],{"className":509},[415,450],[394,511,381],{"className":512},[415,419,450],[394,514,457],{"className":515},[516,450],"mbin",[394,518,520],{"className":519},[415,450],"2",[394,522],{"className":523},[465,524],"allowbreak",[394,526],{"className":527,"style":528},[465],"margin-right:0.4444em;",[394,530,532,536,541,552,556,559],{"className":531},[406],[394,533],{"className":534,"style":535},[410],"height:1em;vertical-align:-0.25em;",[394,537,540],{"className":538},[539],"mopen","(",[394,542,544],{"className":543},[415],[394,545,547],{"className":546},[415],[394,548,551],{"className":549},[415,550],"mathrm","mod",[394,553],{"className":554,"style":555},[465],"margin-right:0.3333em;",[394,557,381],{"className":558},[415,419],[394,560,563],{"className":561},[562],"mclose",")"," are the hinge on\nwhich all of practical combinatorics swings: almost\nevery counting answer is a ratio of factorials, and a ratio modulo a prime is a\nproduct with an inverse. This lesson assembles the counting toolkit (permutations,\ncombinations, Pascal's rule, stars and bars) and then shows how to evaluate those\nquantities ",[389,566,567],{},"modulo a prime"," in constant time after a linear precompute. We finish\nwith two structural principles that recur everywhere: ",[389,570,571],{},"inclusion–exclusion"," for\ncounting unions, and the ",[389,574,575],{},"Chinese Remainder Theorem"," for stitching together\ncongruences.",[578,579,581],"h2",{"id":580},"permutations-and-combinations","Permutations and combinations",[381,583,584,585,588,589,606,607,643],{},"A ",[389,586,587],{},"permutation"," is an ordering of distinct objects. There are ",[394,590,592],{"className":591},[397],[394,593,595],{"className":594,"ariaHidden":402},[401],[394,596,598,602],{"className":597},[406],[394,599],{"className":600,"style":601},[410],"height:0.4306em;",[394,603,605],{"className":604},[415,419],"n"," choices for the\nfirst position, ",[394,608,610],{"className":609},[397],[394,611,613,633],{"className":612,"ariaHidden":402},[401],[394,614,616,620,623,627,630],{"className":615},[406],[394,617],{"className":618,"style":619},[410],"height:0.6667em;vertical-align:-0.0833em;",[394,621,605],{"className":622},[415,419],[394,624],{"className":625,"style":626},[465],"margin-right:0.2222em;",[394,628,457],{"className":629},[516],[394,631],{"className":632,"style":626},[465],[394,634,636,640],{"className":635},[406],[394,637],{"className":638,"style":639},[410],"height:0.6444em;",[394,641,461],{"className":642},[415]," for the second, and so on, giving",[394,645,648],{"className":646},[647],"katex-display",[394,649,651],{"className":650},[397],[394,652,654,678,698,719,755,793],{"className":653,"ariaHidden":402},[401],[394,655,657,661,664,668,671,675],{"className":656},[406],[394,658],{"className":659,"style":660},[410],"height:0.6944em;",[394,662,605],{"className":663},[415,419],[394,665,667],{"className":666},[562],"!",[394,669],{"className":670,"style":466},[465],[394,672,674],{"className":673},[470],"=",[394,676],{"className":677,"style":466},[465],[394,679,681,685,688,691,695],{"className":680},[406],[394,682],{"className":683,"style":684},[410],"height:0.4445em;",[394,686,605],{"className":687},[415,419],[394,689],{"className":690,"style":626},[465],[394,692,694],{"className":693},[516],"⋅",[394,696],{"className":697,"style":626},[465],[394,699,701,704,707,710,713,716],{"className":700},[406],[394,702],{"className":703,"style":535},[410],[394,705,540],{"className":706},[539],[394,708,605],{"className":709},[415,419],[394,711],{"className":712,"style":626},[465],[394,714,457],{"className":715},[516],[394,717],{"className":718,"style":626},[465],[394,720,722,725,728,731,735,740,743,746,749,752],{"className":721},[406],[394,723],{"className":724,"style":535},[410],[394,726,461],{"className":727},[415],[394,729,563],{"className":730},[562],[394,732],{"className":733,"style":734},[465],"margin-right:0.1667em;",[394,736,739],{"className":737},[738],"minner","⋯",[394,741],{"className":742,"style":734},[465],[394,744,520],{"className":745},[415],[394,747],{"className":748,"style":626},[465],[394,750,694],{"className":751},[516],[394,753],{"className":754,"style":626},[465],[394,756,758,762,765,770,774,777,781,784,787,790],{"className":757},[406],[394,759],{"className":760,"style":761},[410],"height:0.8889em;vertical-align:-0.1944em;",[394,763,461],{"className":764},[415],[394,766,769],{"className":767},[768],"mpunct",",",[394,771],{"className":772,"style":773},[465],"margin-right:2em;",[394,775],{"className":776,"style":734},[465],[394,778,780],{"className":779},[415],"0",[394,782,667],{"className":783},[562],[394,785],{"className":786,"style":466},[465],[394,788,674],{"className":789},[470],[394,791],{"className":792,"style":466},[465],[394,794,796,799],{"className":795},[406],[394,797],{"className":798,"style":639},[410],[394,800,802],{"className":801},[415],"1.",[381,804,805,806,823,824,839,840,855],{},"If we order only ",[394,807,809],{"className":808},[397],[394,810,812],{"className":811,"ariaHidden":402},[401],[394,813,815,818],{"className":814},[406],[394,816],{"className":817,"style":601},[410],[394,819,822],{"className":820,"style":821},[415,419],"margin-right:0.0278em;","r"," of the ",[394,825,827],{"className":826},[397],[394,828,830],{"className":829,"ariaHidden":402},[401],[394,831,833,836],{"className":832},[406],[394,834],{"className":835,"style":601},[410],[394,837,605],{"className":838},[415,419]," objects, we stop the product after ",[394,841,843],{"className":842},[397],[394,844,846],{"className":845,"ariaHidden":402},[401],[394,847,849,852],{"className":848},[406],[394,850],{"className":851,"style":601},[410],[394,853,822],{"className":854,"style":821},[415,419]," factors:",[394,857,859],{"className":858},[647],[394,860,862],{"className":861},[397],[394,863,865,892,1005,1029,1065,1084],{"className":864,"ariaHidden":402},[401],[394,866,868,872,875,880,883,886,889],{"className":867},[406],[394,869],{"className":870,"style":871},[410],"height:0.6833em;",[394,873,605],{"className":874},[415,419],[394,876,879],{"className":877,"style":878},[415,419],"margin-right:0.1389em;","P",[394,881,822],{"className":882,"style":821},[415,419],[394,884],{"className":885,"style":466},[465],[394,887,674],{"className":888},[470],[394,890],{"className":891,"style":466},[465],[394,893,895,899,996,999,1002],{"className":894},[406],[394,896],{"className":897,"style":898},[410],"height:2.3074em;vertical-align:-0.936em;",[394,900,902,906,993],{"className":901},[415],[394,903],{"className":904},[539,905],"nulldelimiter",[394,907,910],{"className":908},[909],"mfrac",[394,911,914,984],{"className":912},[427,913],"vlist-t2",[394,915,917,979],{"className":916},[431],[394,918,921,953,964],{"className":919,"style":920},[435],"height:1.3714em;",[394,922,924,928],{"style":923},"top:-2.314em;",[394,925],{"className":926,"style":927},[442],"height:3em;",[394,929,931,934,937,940,943,946,949],{"className":930},[415],[394,932,540],{"className":933},[539],[394,935,605],{"className":936},[415,419],[394,938],{"className":939,"style":626},[465],[394,941,457],{"className":942},[516],[394,944],{"className":945,"style":626},[465],[394,947,822],{"className":948,"style":821},[415,419],[394,950,952],{"className":951},[562],")!",[394,954,956,959],{"style":955},"top:-3.23em;",[394,957],{"className":958,"style":927},[442],[394,960],{"className":961,"style":963},[962],"frac-line","border-bottom-width:0.04em;",[394,965,967,970],{"style":966},"top:-3.677em;",[394,968],{"className":969,"style":927},[442],[394,971,973,976],{"className":972},[415],[394,974,605],{"className":975},[415,419],[394,977,667],{"className":978},[562],[394,980,983],{"className":981},[982],"vlist-s","​",[394,985,987],{"className":986},[431],[394,988,991],{"className":989,"style":990},[435],"height:0.936em;",[394,992],{},[394,994],{"className":995},[562,905],[394,997],{"className":998,"style":466},[465],[394,1000,674],{"className":1001},[470],[394,1003],{"className":1004,"style":466},[465],[394,1006,1008,1011,1014,1017,1020,1023,1026],{"className":1007},[406],[394,1009],{"className":1010,"style":535},[410],[394,1012,605],{"className":1013},[415,419],[394,1015,540],{"className":1016},[539],[394,1018,605],{"className":1019},[415,419],[394,1021],{"className":1022,"style":626},[465],[394,1024,457],{"className":1025},[516],[394,1027],{"className":1028,"style":626},[465],[394,1030,1032,1035,1038,1041,1044,1047,1050,1053,1056,1059,1062],{"className":1031},[406],[394,1033],{"className":1034,"style":535},[410],[394,1036,461],{"className":1037},[415],[394,1039,563],{"className":1040},[562],[394,1042],{"className":1043,"style":734},[465],[394,1045,739],{"className":1046},[738],[394,1048],{"className":1049,"style":734},[465],[394,1051,540],{"className":1052},[539],[394,1054,605],{"className":1055},[415,419],[394,1057],{"className":1058,"style":626},[465],[394,1060,457],{"className":1061},[516],[394,1063],{"className":1064,"style":626},[465],[394,1066,1068,1071,1074,1077,1081],{"className":1067},[406],[394,1069],{"className":1070,"style":619},[410],[394,1072,822],{"className":1073,"style":821},[415,419],[394,1075],{"className":1076,"style":626},[465],[394,1078,1080],{"className":1079},[516],"+",[394,1082],{"className":1083,"style":626},[465],[394,1085,1087,1090,1093,1096],{"className":1086},[406],[394,1088],{"className":1089,"style":535},[410],[394,1091,461],{"className":1092},[415],[394,1094,563],{"className":1095},[562],[394,1097,1099],{"className":1098},[415],".",[381,1101,584,1102,1105,1106,1110,1111,1126,1127,1142,1143,1161,1162,1183,1184,1202],{},[389,1103,1104],{},"combination"," counts ",[1107,1108,1109],"em",{},"subsets"," of size ",[394,1112,1114],{"className":1113},[397],[394,1115,1117],{"className":1116,"ariaHidden":402},[401],[394,1118,1120,1123],{"className":1119},[406],[394,1121],{"className":1122,"style":601},[410],[394,1124,822],{"className":1125,"style":821},[415,419],", where orderings no longer matter.\nEach ",[394,1128,1130],{"className":1129},[397],[394,1131,1133],{"className":1132,"ariaHidden":402},[401],[394,1134,1136,1139],{"className":1135},[406],[394,1137],{"className":1138,"style":601},[410],[394,1140,822],{"className":1141,"style":821},[415,419],"-subset can be ordered in ",[394,1144,1146],{"className":1145},[397],[394,1147,1149],{"className":1148,"ariaHidden":402},[401],[394,1150,1152,1155,1158],{"className":1151},[406],[394,1153],{"className":1154,"style":660},[410],[394,1156,822],{"className":1157,"style":821},[415,419],[394,1159,667],{"className":1160},[562]," ways, so dividing ",[394,1163,1165],{"className":1164},[397],[394,1166,1168],{"className":1167,"ariaHidden":402},[401],[394,1169,1171,1174,1177,1180],{"className":1170},[406],[394,1172],{"className":1173,"style":871},[410],[394,1175,605],{"className":1176},[415,419],[394,1178,879],{"className":1179,"style":878},[415,419],[394,1181,822],{"className":1182,"style":821},[415,419]," by ",[394,1185,1187],{"className":1186},[397],[394,1188,1190],{"className":1189,"ariaHidden":402},[401],[394,1191,1193,1196,1199],{"className":1192},[406],[394,1194],{"className":1195,"style":660},[410],[394,1197,822],{"className":1198,"style":821},[415,419],[394,1200,667],{"className":1201},[562]," removes the\novercount:",[394,1204,1206],{"className":1205},[647],[394,1207,1209],{"className":1208},[397],[394,1210,1212,1293,1381],{"className":1211,"ariaHidden":402},[401],[394,1213,1215,1219,1284,1287,1290],{"className":1214},[406],[394,1216],{"className":1217,"style":1218},[410],"height:2.4em;vertical-align:-0.95em;",[394,1220,1222,1231,1278],{"className":1221},[415],[394,1223,1227],{"className":1224,"style":1226},[539,1225],"delimcenter","top:0em;",[394,1228,540],{"className":1229},[1230,449],"delimsizing",[394,1232,1234],{"className":1233},[909],[394,1235,1237,1269],{"className":1236},[427,913],[394,1238,1240,1266],{"className":1239},[431],[394,1241,1244,1255],{"className":1242,"style":1243},[435],"height:1.1076em;",[394,1245,1246,1249],{"style":923},[394,1247],{"className":1248,"style":927},[442],[394,1250,1252],{"className":1251},[415],[394,1253,822],{"className":1254,"style":821},[415,419],[394,1256,1257,1260],{"style":966},[394,1258],{"className":1259,"style":927},[442],[394,1261,1263],{"className":1262},[415],[394,1264,605],{"className":1265},[415,419],[394,1267,983],{"className":1268},[982],[394,1270,1272],{"className":1271},[431],[394,1273,1276],{"className":1274,"style":1275},[435],"height:0.686em;",[394,1277],{},[394,1279,1281],{"className":1280,"style":1226},[562,1225],[394,1282,563],{"className":1283},[1230,449],[394,1285],{"className":1286,"style":466},[465],[394,1288,674],{"className":1289},[470],[394,1291],{"className":1292,"style":466},[465],[394,1294,1296,1300,1372,1375,1378],{"className":1295},[406],[394,1297],{"className":1298,"style":1299},[410],"height:2.0463em;vertical-align:-0.686em;",[394,1301,1303,1306,1369],{"className":1302},[415],[394,1304],{"className":1305},[539,905],[394,1307,1309],{"className":1308},[909],[394,1310,1312,1361],{"className":1311},[427,913],[394,1313,1315,1358],{"className":1314},[431],[394,1316,1319,1333,1341],{"className":1317,"style":1318},[435],"height:1.3603em;",[394,1320,1321,1324],{"style":923},[394,1322],{"className":1323,"style":927},[442],[394,1325,1327,1330],{"className":1326},[415],[394,1328,822],{"className":1329,"style":821},[415,419],[394,1331,667],{"className":1332},[562],[394,1334,1335,1338],{"style":955},[394,1336],{"className":1337,"style":927},[442],[394,1339],{"className":1340,"style":963},[962],[394,1342,1343,1346],{"style":966},[394,1344],{"className":1345,"style":927},[442],[394,1347,1349,1352,1355],{"className":1348},[415],[394,1350,605],{"className":1351},[415,419],[394,1353,879],{"className":1354,"style":878},[415,419],[394,1356,822],{"className":1357,"style":821},[415,419],[394,1359,983],{"className":1360},[982],[394,1362,1364],{"className":1363},[431],[394,1365,1367],{"className":1366,"style":1275},[435],[394,1368],{},[394,1370],{"className":1371},[562,905],[394,1373],{"className":1374,"style":466},[465],[394,1376,674],{"className":1377},[470],[394,1379],{"className":1380,"style":466},[465],[394,1382,1384,1387,1479],{"className":1383},[406],[394,1385],{"className":1386,"style":898},[410],[394,1388,1390,1393,1476],{"className":1389},[415],[394,1391],{"className":1392},[539,905],[394,1394,1396],{"className":1395},[909],[394,1397,1399,1468],{"className":1398},[427,913],[394,1400,1402,1465],{"className":1401},[431],[394,1403,1405,1443,1451],{"className":1404,"style":920},[435],[394,1406,1407,1410],{"style":923},[394,1408],{"className":1409,"style":927},[442],[394,1411,1413,1416,1419,1422,1425,1428,1431,1434,1437,1440],{"className":1412},[415],[394,1414,822],{"className":1415,"style":821},[415,419],[394,1417,667],{"className":1418},[562],[394,1420],{"className":1421,"style":734},[465],[394,1423,540],{"className":1424},[539],[394,1426,605],{"className":1427},[415,419],[394,1429],{"className":1430,"style":626},[465],[394,1432,457],{"className":1433},[516],[394,1435],{"className":1436,"style":626},[465],[394,1438,822],{"className":1439,"style":821},[415,419],[394,1441,952],{"className":1442},[562],[394,1444,1445,1448],{"style":955},[394,1446],{"className":1447,"style":927},[442],[394,1449],{"className":1450,"style":963},[962],[394,1452,1453,1456],{"style":966},[394,1454],{"className":1455,"style":927},[442],[394,1457,1459,1462],{"className":1458},[415],[394,1460,605],{"className":1461},[415,419],[394,1463,667],{"className":1464},[562],[394,1466,983],{"className":1467},[982],[394,1469,1471],{"className":1470},[431],[394,1472,1474],{"className":1473,"style":990},[435],[394,1475],{},[394,1477],{"className":1478},[562,905],[394,1480,1099],{"className":1481},[415],[381,1483,1484,1485,1569,1570,1604,1605,1099,1608,1617,1618,1785,1786,1801,1802,1835,1836,1099],{},"The quantity ",[394,1486,1488],{"className":1487},[397],[394,1489,1491],{"className":1490,"ariaHidden":402},[401],[394,1492,1494,1498],{"className":1493},[406],[394,1495],{"className":1496,"style":1497},[410],"height:1.2em;vertical-align:-0.35em;",[394,1499,1501,1508,1563],{"className":1500},[415],[394,1502,1504],{"className":1503,"style":1226},[539,1225],[394,1505,540],{"className":1506},[1230,1507],"size1",[394,1509,1511],{"className":1510},[909],[394,1512,1514,1554],{"className":1513},[427,913],[394,1515,1517,1551],{"className":1516},[431],[394,1518,1521,1536],{"className":1519,"style":1520},[435],"height:0.7454em;",[394,1522,1524,1527],{"style":1523},"top:-2.355em;",[394,1525],{"className":1526,"style":443},[442],[394,1528,1530],{"className":1529},[447,448,449,450],[394,1531,1533],{"className":1532},[415,450],[394,1534,822],{"className":1535,"style":821},[415,419,450],[394,1537,1539,1542],{"style":1538},"top:-3.144em;",[394,1540],{"className":1541,"style":443},[442],[394,1543,1545],{"className":1544},[447,448,449,450],[394,1546,1548],{"className":1547},[415,450],[394,1549,605],{"className":1550},[415,419,450],[394,1552,983],{"className":1553},[982],[394,1555,1557],{"className":1556},[431],[394,1558,1561],{"className":1559,"style":1560},[435],"height:0.345em;",[394,1562],{},[394,1564,1566],{"className":1565,"style":1226},[562,1225],[394,1567,563],{"className":1568},[1230,1507],", read ",[1571,1572,1573,1588,1589,769],"q",{},[394,1574,1576],{"className":1575},[397],[394,1577,1579],{"className":1578,"ariaHidden":402},[401],[394,1580,1582,1585],{"className":1581},[406],[394,1583],{"className":1584,"style":601},[410],[394,1586,605],{"className":1587},[415,419]," choose ",[394,1590,1592],{"className":1591},[397],[394,1593,1595],{"className":1594,"ariaHidden":402},[401],[394,1596,1598,1601],{"className":1597},[406],[394,1599],{"className":1600,"style":601},[410],[394,1602,822],{"className":1603,"style":821},[415,419]," is the ",[389,1606,1607],{},"binomial coefficient",[1609,1610,1611],"sup",{},[384,1612,461],{"href":1613,"ariaDescribedBy":1614,"dataFootnoteRef":376,"id":1616},"#user-content-fn-clrs-count",[1615],"footnote-label","user-content-fnref-clrs-count","\nIt is symmetric, ",[394,1619,1621],{"className":1620},[397],[394,1622,1624,1705],{"className":1623,"ariaHidden":402},[401],[394,1625,1627,1630,1696,1699,1702],{"className":1626},[406],[394,1628],{"className":1629,"style":1497},[410],[394,1631,1633,1639,1690],{"className":1632},[415],[394,1634,1636],{"className":1635,"style":1226},[539,1225],[394,1637,540],{"className":1638},[1230,1507],[394,1640,1642],{"className":1641},[909],[394,1643,1645,1682],{"className":1644},[427,913],[394,1646,1648,1679],{"className":1647},[431],[394,1649,1651,1665],{"className":1650,"style":1520},[435],[394,1652,1653,1656],{"style":1523},[394,1654],{"className":1655,"style":443},[442],[394,1657,1659],{"className":1658},[447,448,449,450],[394,1660,1662],{"className":1661},[415,450],[394,1663,822],{"className":1664,"style":821},[415,419,450],[394,1666,1667,1670],{"style":1538},[394,1668],{"className":1669,"style":443},[442],[394,1671,1673],{"className":1672},[447,448,449,450],[394,1674,1676],{"className":1675},[415,450],[394,1677,605],{"className":1678},[415,419,450],[394,1680,983],{"className":1681},[982],[394,1683,1685],{"className":1684},[431],[394,1686,1688],{"className":1687,"style":1560},[435],[394,1689],{},[394,1691,1693],{"className":1692,"style":1226},[562,1225],[394,1694,563],{"className":1695},[1230,1507],[394,1697],{"className":1698,"style":466},[465],[394,1700,674],{"className":1701},[470],[394,1703],{"className":1704,"style":466},[465],[394,1706,1708,1712],{"className":1707},[406],[394,1709],{"className":1710,"style":1711},[410],"height:1.2533em;vertical-align:-0.4033em;",[394,1713,1715,1721,1779],{"className":1714},[415],[394,1716,1718],{"className":1717,"style":1226},[539,1225],[394,1719,540],{"className":1720},[1230,1507],[394,1722,1724],{"className":1723},[909],[394,1725,1727,1770],{"className":1726},[427,913],[394,1728,1730,1767],{"className":1729},[431],[394,1731,1733,1753],{"className":1732,"style":1520},[435],[394,1734,1735,1738],{"style":1523},[394,1736],{"className":1737,"style":443},[442],[394,1739,1741],{"className":1740},[447,448,449,450],[394,1742,1744,1747,1750],{"className":1743},[415,450],[394,1745,605],{"className":1746},[415,419,450],[394,1748,457],{"className":1749},[516,450],[394,1751,822],{"className":1752,"style":821},[415,419,450],[394,1754,1755,1758],{"style":1538},[394,1756],{"className":1757,"style":443},[442],[394,1759,1761],{"className":1760},[447,448,449,450],[394,1762,1764],{"className":1763},[415,450],[394,1765,605],{"className":1766},[415,419,450],[394,1768,983],{"className":1769},[982],[394,1771,1773],{"className":1772},[431],[394,1774,1777],{"className":1775,"style":1776},[435],"height:0.4033em;",[394,1778],{},[394,1780,1782],{"className":1781,"style":1226},[562,1225],[394,1783,563],{"className":1784},[1230,1507]," (choosing which ",[394,1787,1789],{"className":1788},[397],[394,1790,1792],{"className":1791,"ariaHidden":402},[401],[394,1793,1795,1798],{"className":1794},[406],[394,1796],{"className":1797,"style":601},[410],[394,1799,822],{"className":1800,"style":821},[415,419]," to include is the\nsame as choosing which ",[394,1803,1805],{"className":1804},[397],[394,1806,1808,1826],{"className":1807,"ariaHidden":402},[401],[394,1809,1811,1814,1817,1820,1823],{"className":1810},[406],[394,1812],{"className":1813,"style":619},[410],[394,1815,605],{"className":1816},[415,419],[394,1818],{"className":1819,"style":626},[465],[394,1821,457],{"className":1822},[516],[394,1824],{"className":1825,"style":626},[465],[394,1827,1829,1832],{"className":1828},[406],[394,1830],{"className":1831,"style":601},[410],[394,1833,822],{"className":1834,"style":821},[415,419]," to exclude), and the two boundary values are\n",[394,1837,1839],{"className":1838},[397],[394,1840,1842,1923,2004],{"className":1841,"ariaHidden":402},[401],[394,1843,1845,1848,1914,1917,1920],{"className":1844},[406],[394,1846],{"className":1847,"style":1497},[410],[394,1849,1851,1857,1908],{"className":1850},[415],[394,1852,1854],{"className":1853,"style":1226},[539,1225],[394,1855,540],{"className":1856},[1230,1507],[394,1858,1860],{"className":1859},[909],[394,1861,1863,1900],{"className":1862},[427,913],[394,1864,1866,1897],{"className":1865},[431],[394,1867,1869,1883],{"className":1868,"style":1520},[435],[394,1870,1871,1874],{"style":1523},[394,1872],{"className":1873,"style":443},[442],[394,1875,1877],{"className":1876},[447,448,449,450],[394,1878,1880],{"className":1879},[415,450],[394,1881,780],{"className":1882},[415,450],[394,1884,1885,1888],{"style":1538},[394,1886],{"className":1887,"style":443},[442],[394,1889,1891],{"className":1890},[447,448,449,450],[394,1892,1894],{"className":1893},[415,450],[394,1895,605],{"className":1896},[415,419,450],[394,1898,983],{"className":1899},[982],[394,1901,1903],{"className":1902},[431],[394,1904,1906],{"className":1905,"style":1560},[435],[394,1907],{},[394,1909,1911],{"className":1910,"style":1226},[562,1225],[394,1912,563],{"className":1913},[1230,1507],[394,1915],{"className":1916,"style":466},[465],[394,1918,674],{"className":1919},[470],[394,1921],{"className":1922,"style":466},[465],[394,1924,1926,1929,1995,1998,2001],{"className":1925},[406],[394,1927],{"className":1928,"style":1497},[410],[394,1930,1932,1938,1989],{"className":1931},[415],[394,1933,1935],{"className":1934,"style":1226},[539,1225],[394,1936,540],{"className":1937},[1230,1507],[394,1939,1941],{"className":1940},[909],[394,1942,1944,1981],{"className":1943},[427,913],[394,1945,1947,1978],{"className":1946},[431],[394,1948,1950,1964],{"className":1949,"style":1520},[435],[394,1951,1952,1955],{"style":1523},[394,1953],{"className":1954,"style":443},[442],[394,1956,1958],{"className":1957},[447,448,449,450],[394,1959,1961],{"className":1960},[415,450],[394,1962,605],{"className":1963},[415,419,450],[394,1965,1966,1969],{"style":1538},[394,1967],{"className":1968,"style":443},[442],[394,1970,1972],{"className":1971},[447,448,449,450],[394,1973,1975],{"className":1974},[415,450],[394,1976,605],{"className":1977},[415,419,450],[394,1979,983],{"className":1980},[982],[394,1982,1984],{"className":1983},[431],[394,1985,1987],{"className":1986,"style":1560},[435],[394,1988],{},[394,1990,1992],{"className":1991,"style":1226},[562,1225],[394,1993,563],{"className":1994},[1230,1507],[394,1996],{"className":1997,"style":466},[465],[394,1999,674],{"className":2000},[470],[394,2002],{"className":2003,"style":466},[465],[394,2005,2007,2010],{"className":2006},[406],[394,2008],{"className":2009,"style":639},[410],[394,2011,461],{"className":2012},[415],[2014,2015,2017],"h3",{"id":2016},"pascals-rule","Pascal's rule",[381,2019,2020],{},"Binomial coefficients satisfy a recurrence that lets us build them additively, with\nno division at all:",[2022,2023,2025],"callout",{"type":2024},"lemma",[381,2026,2027,2030,2031,2087,2088],{},[389,2028,2029],{},"Lemma (Pascal's rule)."," For ",[394,2032,2034],{"className":2033},[397],[394,2035,2037,2057,2078],{"className":2036,"ariaHidden":402},[401],[394,2038,2040,2044,2047,2050,2054],{"className":2039},[406],[394,2041],{"className":2042,"style":2043},[410],"height:0.6835em;vertical-align:-0.0391em;",[394,2045,780],{"className":2046},[415],[394,2048],{"className":2049,"style":466},[465],[394,2051,2053],{"className":2052},[470],"\u003C",[394,2055],{"className":2056,"style":466},[465],[394,2058,2060,2064,2069,2072,2075],{"className":2059},[406],[394,2061],{"className":2062,"style":2063},[410],"height:0.7335em;vertical-align:-0.0391em;",[394,2065,2068],{"className":2066,"style":2067},[415,419],"margin-right:0.0315em;","k",[394,2070],{"className":2071,"style":466},[465],[394,2073,2053],{"className":2074},[470],[394,2076],{"className":2077,"style":466},[465],[394,2079,2081,2084],{"className":2080},[406],[394,2082],{"className":2083,"style":601},[410],[394,2085,605],{"className":2086},[415,419],",\n",[394,2089,2091],{"className":2090},[397],[394,2092,2094,2175,2270],{"className":2093,"ariaHidden":402},[401],[394,2095,2097,2100,2166,2169,2172],{"className":2096},[406],[394,2098],{"className":2099,"style":1497},[410],[394,2101,2103,2109,2160],{"className":2102},[415],[394,2104,2106],{"className":2105,"style":1226},[539,1225],[394,2107,540],{"className":2108},[1230,1507],[394,2110,2112],{"className":2111},[909],[394,2113,2115,2152],{"className":2114},[427,913],[394,2116,2118,2149],{"className":2117},[431],[394,2119,2121,2135],{"className":2120,"style":1520},[435],[394,2122,2123,2126],{"style":1523},[394,2124],{"className":2125,"style":443},[442],[394,2127,2129],{"className":2128},[447,448,449,450],[394,2130,2132],{"className":2131},[415,450],[394,2133,2068],{"className":2134,"style":2067},[415,419,450],[394,2136,2137,2140],{"style":1538},[394,2138],{"className":2139,"style":443},[442],[394,2141,2143],{"className":2142},[447,448,449,450],[394,2144,2146],{"className":2145},[415,450],[394,2147,605],{"className":2148},[415,419,450],[394,2150,983],{"className":2151},[982],[394,2153,2155],{"className":2154},[431],[394,2156,2158],{"className":2157,"style":1560},[435],[394,2159],{},[394,2161,2163],{"className":2162,"style":1226},[562,1225],[394,2164,563],{"className":2165},[1230,1507],[394,2167],{"className":2168,"style":466},[465],[394,2170,674],{"className":2171},[470],[394,2173],{"className":2174,"style":466},[465],[394,2176,2178,2182,2261,2264,2267],{"className":2177},[406],[394,2179],{"className":2180,"style":2181},[410],"height:1.2984em;vertical-align:-0.4033em;",[394,2183,2185,2191,2255],{"className":2184},[415],[394,2186,2188],{"className":2187,"style":1226},[539,1225],[394,2189,540],{"className":2190},[1230,1507],[394,2192,2194],{"className":2193},[909],[394,2195,2197,2247],{"className":2196},[427,913],[394,2198,2200,2244],{"className":2199},[431],[394,2201,2204,2224],{"className":2202,"style":2203},[435],"height:0.8951em;",[394,2205,2206,2209],{"style":1523},[394,2207],{"className":2208,"style":443},[442],[394,2210,2212],{"className":2211},[447,448,449,450],[394,2213,2215,2218,2221],{"className":2214},[415,450],[394,2216,2068],{"className":2217,"style":2067},[415,419,450],[394,2219,457],{"className":2220},[516,450],[394,2222,461],{"className":2223},[415,450],[394,2225,2226,2229],{"style":1538},[394,2227],{"className":2228,"style":443},[442],[394,2230,2232],{"className":2231},[447,448,449,450],[394,2233,2235,2238,2241],{"className":2234},[415,450],[394,2236,605],{"className":2237},[415,419,450],[394,2239,457],{"className":2240},[516,450],[394,2242,461],{"className":2243},[415,450],[394,2245,983],{"className":2246},[982],[394,2248,2250],{"className":2249},[431],[394,2251,2253],{"className":2252,"style":1776},[435],[394,2254],{},[394,2256,2258],{"className":2257,"style":1226},[562,1225],[394,2259,563],{"className":2260},[1230,1507],[394,2262],{"className":2263,"style":626},[465],[394,2265,1080],{"className":2266},[516],[394,2268],{"className":2269,"style":626},[465],[394,2271,2273,2277,2349],{"className":2272},[406],[394,2274],{"className":2275,"style":2276},[410],"height:1.2451em;vertical-align:-0.35em;",[394,2278,2280,2286,2343],{"className":2279},[415],[394,2281,2283],{"className":2282,"style":1226},[539,1225],[394,2284,540],{"className":2285},[1230,1507],[394,2287,2289],{"className":2288},[909],[394,2290,2292,2335],{"className":2291},[427,913],[394,2293,2295,2332],{"className":2294},[431],[394,2296,2298,2312],{"className":2297,"style":2203},[435],[394,2299,2300,2303],{"style":1523},[394,2301],{"className":2302,"style":443},[442],[394,2304,2306],{"className":2305},[447,448,449,450],[394,2307,2309],{"className":2308},[415,450],[394,2310,2068],{"className":2311,"style":2067},[415,419,450],[394,2313,2314,2317],{"style":1538},[394,2315],{"className":2316,"style":443},[442],[394,2318,2320],{"className":2319},[447,448,449,450],[394,2321,2323,2326,2329],{"className":2322},[415,450],[394,2324,605],{"className":2325},[415,419,450],[394,2327,457],{"className":2328},[516,450],[394,2330,461],{"className":2331},[415,450],[394,2333,983],{"className":2334},[982],[394,2336,2338],{"className":2337},[431],[394,2339,2341],{"className":2340,"style":1560},[435],[394,2342],{},[394,2344,2346],{"className":2345,"style":1226},[562,1225],[394,2347,563],{"className":2348},[1230,1507],[394,2350,1099],{"className":2351},[415],[381,2353,2354,2357,2358,2373,2374,2389,2390,2435,2436,392,2439,2454,2455,2458,2459,2474,2475,2509,2510,2543,2544,2634,2635,2650,2651,2666,2667,2700,2701,2785,2786],{},[1107,2355,2356],{},"Combinatorial proof."," Fix a distinguished element ",[394,2359,2361],{"className":2360},[397],[394,2362,2364],{"className":2363,"ariaHidden":402},[401],[394,2365,2367,2370],{"className":2366},[406],[394,2368],{"className":2369,"style":601},[410],[394,2371,605],{"className":2372},[415,419],". Every ",[394,2375,2377],{"className":2376},[397],[394,2378,2380],{"className":2379,"ariaHidden":402},[401],[394,2381,2383,2386],{"className":2382},[406],[394,2384],{"className":2385,"style":660},[410],[394,2387,2068],{"className":2388,"style":2067},[415,419],"-subset of\n",[394,2391,2393],{"className":2392},[397],[394,2394,2396],{"className":2395,"ariaHidden":402},[401],[394,2397,2399,2402,2406,2409,2412,2415,2419,2422,2425,2428,2431],{"className":2398},[406],[394,2400],{"className":2401,"style":535},[410],[394,2403,2405],{"className":2404},[539],"{",[394,2407,461],{"className":2408},[415],[394,2410,769],{"className":2411},[768],[394,2413],{"className":2414,"style":734},[465],[394,2416,2418],{"className":2417},[738],"…",[394,2420],{"className":2421,"style":734},[465],[394,2423,769],{"className":2424},[768],[394,2426],{"className":2427,"style":734},[465],[394,2429,605],{"className":2430},[415,419],[394,2432,2434],{"className":2433},[562],"}"," either ",[389,2437,2438],{},"contains",[394,2440,2442],{"className":2441},[397],[394,2443,2445],{"className":2444,"ariaHidden":402},[401],[394,2446,2448,2451],{"className":2447},[406],[394,2449],{"className":2450,"style":601},[410],[394,2452,605],{"className":2453},[415,419]," or ",[389,2456,2457],{},"does not",". Those that contain ",[394,2460,2462],{"className":2461},[397],[394,2463,2465],{"className":2464,"ariaHidden":402},[401],[394,2466,2468,2471],{"className":2467},[406],[394,2469],{"className":2470,"style":601},[410],[394,2472,605],{"className":2473},[415,419]," are\nformed by choosing the remaining ",[394,2476,2478],{"className":2477},[397],[394,2479,2481,2500],{"className":2480,"ariaHidden":402},[401],[394,2482,2484,2488,2491,2494,2497],{"className":2483},[406],[394,2485],{"className":2486,"style":2487},[410],"height:0.7778em;vertical-align:-0.0833em;",[394,2489,2068],{"className":2490,"style":2067},[415,419],[394,2492],{"className":2493,"style":626},[465],[394,2495,457],{"className":2496},[516],[394,2498],{"className":2499,"style":626},[465],[394,2501,2503,2506],{"className":2502},[406],[394,2504],{"className":2505,"style":639},[410],[394,2507,461],{"className":2508},[415]," elements from the other ",[394,2511,2513],{"className":2512},[397],[394,2514,2516,2534],{"className":2515,"ariaHidden":402},[401],[394,2517,2519,2522,2525,2528,2531],{"className":2518},[406],[394,2520],{"className":2521,"style":619},[410],[394,2523,605],{"className":2524},[415,419],[394,2526],{"className":2527,"style":626},[465],[394,2529,457],{"className":2530},[516],[394,2532],{"className":2533,"style":626},[465],[394,2535,2537,2540],{"className":2536},[406],[394,2538],{"className":2539,"style":639},[410],[394,2541,461],{"className":2542},[415],", giving\n",[394,2545,2547],{"className":2546},[397],[394,2548,2550],{"className":2549,"ariaHidden":402},[401],[394,2551,2553,2556],{"className":2552},[406],[394,2554],{"className":2555,"style":2181},[410],[394,2557,2559,2565,2628],{"className":2558},[415],[394,2560,2562],{"className":2561,"style":1226},[539,1225],[394,2563,540],{"className":2564},[1230,1507],[394,2566,2568],{"className":2567},[909],[394,2569,2571,2620],{"className":2570},[427,913],[394,2572,2574,2617],{"className":2573},[431],[394,2575,2577,2597],{"className":2576,"style":2203},[435],[394,2578,2579,2582],{"style":1523},[394,2580],{"className":2581,"style":443},[442],[394,2583,2585],{"className":2584},[447,448,449,450],[394,2586,2588,2591,2594],{"className":2587},[415,450],[394,2589,2068],{"className":2590,"style":2067},[415,419,450],[394,2592,457],{"className":2593},[516,450],[394,2595,461],{"className":2596},[415,450],[394,2598,2599,2602],{"style":1538},[394,2600],{"className":2601,"style":443},[442],[394,2603,2605],{"className":2604},[447,448,449,450],[394,2606,2608,2611,2614],{"className":2607},[415,450],[394,2609,605],{"className":2610},[415,419,450],[394,2612,457],{"className":2613},[516,450],[394,2615,461],{"className":2616},[415,450],[394,2618,983],{"className":2619},[982],[394,2621,2623],{"className":2622},[431],[394,2624,2626],{"className":2625,"style":1776},[435],[394,2627],{},[394,2629,2631],{"className":2630,"style":1226},[562,1225],[394,2632,563],{"className":2633},[1230,1507]," of them. Those that omit ",[394,2636,2638],{"className":2637},[397],[394,2639,2641],{"className":2640,"ariaHidden":402},[401],[394,2642,2644,2647],{"className":2643},[406],[394,2645],{"className":2646,"style":601},[410],[394,2648,605],{"className":2649},[415,419]," choose all ",[394,2652,2654],{"className":2653},[397],[394,2655,2657],{"className":2656,"ariaHidden":402},[401],[394,2658,2660,2663],{"className":2659},[406],[394,2661],{"className":2662,"style":660},[410],[394,2664,2068],{"className":2665,"style":2067},[415,419]," elements from the\nother ",[394,2668,2670],{"className":2669},[397],[394,2671,2673,2691],{"className":2672,"ariaHidden":402},[401],[394,2674,2676,2679,2682,2685,2688],{"className":2675},[406],[394,2677],{"className":2678,"style":619},[410],[394,2680,605],{"className":2681},[415,419],[394,2683],{"className":2684,"style":626},[465],[394,2686,457],{"className":2687},[516],[394,2689],{"className":2690,"style":626},[465],[394,2692,2694,2697],{"className":2693},[406],[394,2695],{"className":2696,"style":639},[410],[394,2698,461],{"className":2699},[415],", giving ",[394,2702,2704],{"className":2703},[397],[394,2705,2707],{"className":2706,"ariaHidden":402},[401],[394,2708,2710,2713],{"className":2709},[406],[394,2711],{"className":2712,"style":2276},[410],[394,2714,2716,2722,2779],{"className":2715},[415],[394,2717,2719],{"className":2718,"style":1226},[539,1225],[394,2720,540],{"className":2721},[1230,1507],[394,2723,2725],{"className":2724},[909],[394,2726,2728,2771],{"className":2727},[427,913],[394,2729,2731,2768],{"className":2730},[431],[394,2732,2734,2748],{"className":2733,"style":2203},[435],[394,2735,2736,2739],{"style":1523},[394,2737],{"className":2738,"style":443},[442],[394,2740,2742],{"className":2741},[447,448,449,450],[394,2743,2745],{"className":2744},[415,450],[394,2746,2068],{"className":2747,"style":2067},[415,419,450],[394,2749,2750,2753],{"style":1538},[394,2751],{"className":2752,"style":443},[442],[394,2754,2756],{"className":2755},[447,448,449,450],[394,2757,2759,2762,2765],{"className":2758},[415,450],[394,2760,605],{"className":2761},[415,419,450],[394,2763,457],{"className":2764},[516,450],[394,2766,461],{"className":2767},[415,450],[394,2769,983],{"className":2770},[982],[394,2772,2774],{"className":2773},[431],[394,2775,2777],{"className":2776,"style":1560},[435],[394,2778],{},[394,2780,2782],{"className":2781,"style":1226},[562,1225],[394,2783,563],{"className":2784},[1230,1507],". The two cases are disjoint and exhaustive,\nso their counts add. ",[394,2787,2789],{"className":2788},[397],[394,2790,2792],{"className":2791,"ariaHidden":402},[401],[394,2793,2795,2799],{"className":2794},[406],[394,2796],{"className":2797,"style":2798},[410],"height:0.675em;",[394,2800,2804],{"className":2801},[2802,2803],"enclosing","qed",[394,2805,2808],{"className":2806},[415,2807],"amsrm","□",[381,2810,2811,2812,2815],{},"Arranging these values in rows is ",[389,2813,2814],{},"Pascal's triangle",": each interior entry is the\nsum of the two directly above it.",[2817,2818,2822,3002],"figure",{"className":2819},[2820,2821],"tikz-figure","tikz-diagram-rendered",[2823,2824,2829],"svg",{"xmlns":2825,"width":2826,"height":2827,"viewBox":2828},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","214.695","221.514","-75 -75 161.021 166.135",[2830,2831,2834,2839,2848,2851,2857,2860,2866,2869,2875,2878,2885,2888,2894,2897,2903,2906,2913,2916,2922,2925,2931,2934,2940,2943,2950,2965,2968,2974,2977,2983,2994],"g",{"stroke":2832,"style":2833},"currentColor","stroke-miterlimit:10;stroke-width:.4",[2835,2836],"path",{"fill":2837,"d":2838},"none","M19.955-60.689c0-6.286-5.096-11.381-11.381-11.381S-2.807-66.975-2.807-60.689s5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2840,2842],{"transform":2841},"translate(-2.312 2.9)",[2835,2843],{"d":2844,"fill":2832,"stroke":2832,"className":2845,"style":2847},"M12.481-60.689L9.449-60.689L9.449-61.005Q10.600-61.005 10.600-61.300L10.600-66.024Q10.112-65.791 9.391-65.791L9.391-66.107Q10.521-66.107 11.083-66.683L11.228-66.683Q11.263-66.683 11.296-66.650Q11.329-66.617 11.329-66.582L11.329-61.300Q11.329-61.005 12.481-61.005",[2846],"tikz-text","stroke-width:0.270",[2835,2849],{"fill":2837,"d":2850},"M4.306-26.546c0-6.285-5.096-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381S4.307-20.26 4.307-26.545Zm-11.381 0",[2830,2852,2854],{"transform":2853},"translate(-17.961 37.043)",[2835,2855],{"d":2844,"fill":2832,"stroke":2832,"className":2856,"style":2847},[2846],[2835,2858],{"fill":2837,"d":2859},"M35.604-26.546c0-6.285-5.096-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2861,2863],{"transform":2862},"translate(13.337 37.043)",[2835,2864],{"d":2844,"fill":2832,"stroke":2832,"className":2865,"style":2847},[2846],[2835,2867],{"fill":2837,"d":2868},"M-11.343 7.598c0-6.286-5.096-11.381-11.381-11.381s-11.381 5.095-11.381 11.38 5.095 11.382 11.38 11.382 11.382-5.096 11.382-11.381Zm-11.381 0",[2830,2870,2872],{"transform":2871},"translate(-33.61 71.187)",[2835,2873],{"d":2844,"fill":2832,"stroke":2832,"className":2874,"style":2847},[2846],[2835,2876],{"fill":2837,"d":2877},"M19.955 7.598c0-6.286-5.096-11.381-11.381-11.381S-2.807 1.312-2.807 7.597 2.288 18.98 8.573 18.98s11.382-5.096 11.382-11.381Zm-11.381 0",[2830,2879,2881],{"transform":2880},"translate(-2.312 71.187)",[2835,2882],{"d":2883,"fill":2832,"stroke":2832,"className":2884,"style":2847},"M12.481-60.689L9.031-60.689L9.031-60.922Q9.031-60.935 9.062-60.966L10.516-62.543Q10.982-63.040 11.235-63.345Q11.488-63.651 11.679-64.062Q11.870-64.473 11.870-64.912Q11.870-65.501 11.547-65.934Q11.224-66.367 10.644-66.367Q10.380-66.367 10.134-66.257Q9.888-66.147 9.712-65.960Q9.536-65.773 9.440-65.523L9.519-65.523Q9.721-65.523 9.864-65.387Q10.007-65.251 10.007-65.035Q10.007-64.829 9.864-64.690Q9.721-64.552 9.519-64.552Q9.317-64.552 9.174-64.695Q9.031-64.837 9.031-65.035Q9.031-65.497 9.268-65.870Q9.506-66.244 9.906-66.463Q10.305-66.683 10.754-66.683Q11.277-66.683 11.731-66.468Q12.186-66.252 12.459-65.853Q12.731-65.453 12.731-64.912Q12.731-64.517 12.560-64.163Q12.388-63.809 12.123-63.530Q11.857-63.251 11.406-62.866Q10.956-62.482 10.877-62.407L9.853-61.445L10.670-61.445Q11.321-61.445 11.758-61.456Q12.195-61.467 12.226-61.489Q12.296-61.572 12.351-61.812Q12.406-62.051 12.446-62.319L12.731-62.319",[2846],[2835,2886],{"fill":2837,"d":2887},"M51.253 7.598c0-6.286-5.096-11.381-11.381-11.381S28.49 1.312 28.49 7.597 33.586 18.98 39.87 18.98s11.382-5.096 11.382-11.381Zm-11.381 0",[2830,2889,2891],{"transform":2890},"translate(28.986 71.187)",[2835,2892],{"d":2844,"fill":2832,"stroke":2832,"className":2893,"style":2847},[2846],[2835,2895],{"fill":2837,"d":2896},"M-26.992 41.741c0-6.286-5.096-11.381-11.381-11.381s-11.381 5.095-11.381 11.381 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.381Zm-11.381 0",[2830,2898,2900],{"transform":2899},"translate(-49.26 105.33)",[2835,2901],{"d":2844,"fill":2832,"stroke":2832,"className":2902,"style":2847},[2846],[2835,2904],{"fill":2837,"d":2905},"M4.306 41.741c0-6.286-5.096-11.381-11.381-11.381s-11.381 5.095-11.381 11.381 5.095 11.381 11.38 11.381S4.307 48.027 4.307 41.741Zm-11.381 0",[2830,2907,2909],{"transform":2908},"translate(-17.961 105.33)",[2835,2910],{"d":2911,"fill":2832,"stroke":2832,"className":2912,"style":2847},"M9.475-61.410L9.431-61.410Q9.633-61.093 10.020-60.935Q10.407-60.777 10.833-60.777Q11.369-60.777 11.608-61.212Q11.848-61.647 11.848-62.227Q11.848-62.807 11.602-63.247Q11.356-63.686 10.824-63.686L10.204-63.686Q10.178-63.686 10.145-63.715Q10.112-63.743 10.112-63.765L10.112-63.866Q10.112-63.897 10.141-63.921Q10.169-63.945 10.204-63.945L10.723-63.985Q11.189-63.985 11.435-64.457Q11.681-64.930 11.681-65.448Q11.681-65.875 11.468-66.149Q11.255-66.424 10.833-66.424Q10.490-66.424 10.165-66.294Q9.840-66.165 9.655-65.910L9.681-65.910Q9.884-65.910 10.020-65.769Q10.156-65.628 10.156-65.431Q10.156-65.233 10.022-65.099Q9.888-64.965 9.690-64.965Q9.488-64.965 9.350-65.099Q9.211-65.233 9.211-65.431Q9.211-66.020 9.714-66.351Q10.218-66.683 10.833-66.683Q11.211-66.683 11.613-66.543Q12.015-66.402 12.283-66.123Q12.551-65.844 12.551-65.448Q12.551-64.899 12.197-64.462Q11.844-64.024 11.303-63.840Q11.694-63.761 12.039-63.537Q12.384-63.313 12.595-62.972Q12.806-62.631 12.806-62.236Q12.806-61.854 12.643-61.531Q12.481-61.208 12.189-60.972Q11.896-60.737 11.549-60.614Q11.202-60.491 10.833-60.491Q10.385-60.491 9.954-60.652Q9.523-60.812 9.242-61.139Q8.961-61.467 8.961-61.924Q8.961-62.139 9.108-62.282Q9.255-62.425 9.475-62.425Q9.686-62.425 9.831-62.280Q9.976-62.135 9.976-61.924Q9.976-61.713 9.829-61.561Q9.681-61.410 9.475-61.410",[2846],[2835,2914],{"fill":2837,"d":2915},"M35.604 41.741c0-6.286-5.096-11.381-11.381-11.381S12.842 35.455 12.842 41.74s5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.381Zm-11.381 0",[2830,2917,2919],{"transform":2918},"translate(13.337 105.33)",[2835,2920],{"d":2911,"fill":2832,"stroke":2832,"className":2921,"style":2847},[2846],[2835,2923],{"fill":2837,"d":2924},"M66.902 41.741c0-6.286-5.096-11.381-11.381-11.381S44.14 35.455 44.14 41.74s5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.381Zm-11.381 0",[2830,2926,2928],{"transform":2927},"translate(44.635 105.33)",[2835,2929],{"d":2844,"fill":2832,"stroke":2832,"className":2930,"style":2847},[2846],[2835,2932],{"fill":2837,"d":2933},"M-42.641 75.884c0-6.285-5.096-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2935,2937],{"transform":2936},"translate(-64.909 139.473)",[2835,2938],{"d":2844,"fill":2832,"stroke":2832,"className":2939,"style":2847},[2846],[2835,2941],{"fill":2837,"d":2942},"M-11.343 75.884c0-6.285-5.096-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2944,2946],{"transform":2945},"translate(-33.61 139.473)",[2835,2947],{"d":2948,"fill":2832,"stroke":2832,"className":2949,"style":2847},"M11.272-62.166L8.833-62.166L8.833-62.482L11.659-66.630Q11.703-66.683 11.769-66.683L11.923-66.683Q11.962-66.683 11.995-66.650Q12.028-66.617 12.028-66.573L12.028-62.482L12.929-62.482L12.929-62.166L12.028-62.166L12.028-61.300Q12.028-61.005 12.929-61.005L12.929-60.689L10.376-60.689L10.376-61.005Q10.736-61.005 11.004-61.060Q11.272-61.115 11.272-61.300L11.272-62.166M11.329-65.655L9.167-62.482L11.329-62.482",[2846],[2830,2951,2955,2958],{"fill":2952,"stroke":2953,"style":2954},"var(--tk-soft-accent)","var(--tk-accent)","stroke-width:1.2",[2835,2956],{"d":2957},"M19.955 75.884c0-6.285-5.096-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2959,2961],{"transform":2960},"translate(-2.312 139.473)",[2835,2962],{"d":2963,"fill":2832,"stroke":2832,"className":2964,"style":2847},"M10.886-60.491Q10.152-60.491 9.721-60.972Q9.290-61.454 9.126-62.146Q8.961-62.838 8.961-63.585Q8.961-64.314 9.253-65.037Q9.545-65.760 10.099-66.222Q10.653-66.683 11.400-66.683Q11.896-66.683 12.232-66.417Q12.569-66.151 12.569-65.668Q12.569-65.488 12.441-65.360Q12.314-65.233 12.138-65.233Q11.958-65.233 11.828-65.358Q11.699-65.483 11.699-65.668Q11.699-65.782 11.756-65.886Q11.813-65.989 11.914-66.048Q12.015-66.107 12.138-66.107Q12.142-66.107 12.147-66.105Q12.151-66.103 12.156-66.099Q12.041-66.266 11.833-66.345Q11.624-66.424 11.400-66.424Q10.956-66.424 10.598-66.123Q10.240-65.822 10.051-65.369Q9.818-64.763 9.818-63.730Q9.989-64.095 10.290-64.323Q10.591-64.552 10.978-64.552Q11.382-64.552 11.727-64.385Q12.072-64.218 12.309-63.937Q12.547-63.655 12.676-63.293Q12.806-62.930 12.806-62.526Q12.806-61.981 12.562-61.515Q12.318-61.049 11.879-60.770Q11.439-60.491 10.886-60.491M10.886-60.777Q11.347-60.777 11.582-61.034Q11.817-61.291 11.883-61.665Q11.949-62.038 11.949-62.508L11.949-62.543Q11.949-63.031 11.892-63.396Q11.835-63.761 11.606-64.024Q11.378-64.288 10.934-64.288Q10.565-64.288 10.314-64.044Q10.064-63.800 9.949-63.436Q9.835-63.071 9.835-62.724Q9.835-62.605 9.844-62.543Q9.844-62.526 9.842-62.515Q9.840-62.504 9.835-62.491Q9.835-61.840 10.073-61.309Q10.310-60.777 10.886-60.777",[2846],[2835,2966],{"fill":2837,"d":2967},"M51.253 75.884c0-6.285-5.096-11.38-11.381-11.38S28.49 69.598 28.49 75.883s5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[2830,2969,2971],{"transform":2970},"translate(28.986 139.473)",[2835,2972],{"d":2948,"fill":2832,"stroke":2832,"className":2973,"style":2847},[2846],[2835,2975],{"fill":2837,"d":2976},"M82.55 75.884c0-6.285-5.095-11.38-11.38-11.38s-11.381 5.095-11.381 11.38 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.38 0",[2830,2978,2980],{"transform":2979},"translate(60.284 139.473)",[2835,2981],{"d":2844,"fill":2832,"stroke":2832,"className":2982,"style":2847},[2846],[2830,2984,2987,2990],{"fill":2985,"stroke":2985,"style":2986},"var(--tk-warn)","stroke-width:.8",[2835,2988],{"fill":2837,"d":2989},"m-2.25 52.268 5.082 11.089",[2835,2991],{"fill":2837,"d":2992,"style":2993},"M4.77 60.092C3.264 61.298 2.894 62.614 3 63.721c-.769-.803-2.007-1.382-3.905-1.027","stroke-linecap:round;stroke-linejoin:round;stroke-width:.799968",[2830,2995,2996,2999],{"fill":2985,"stroke":2985,"style":2986},[2835,2997],{"fill":2837,"d":2998},"m19.398 52.268-5.083 11.089",[2835,3000],{"fill":2837,"d":3001,"style":2993},"M18.054 62.694c-1.898-.355-3.137.224-3.905 1.027.106-1.107-.264-2.423-1.772-3.629",[3003,3004,3007,3008],"figcaption",{"className":3005},[3006],"tikz-cap","Pascal's triangle — each cell ",[394,3009,3011],{"className":3010},[397],[394,3012,3014,3095,3188],{"className":3013,"ariaHidden":402},[401],[394,3015,3017,3020,3086,3089,3092],{"className":3016},[406],[394,3018],{"className":3019,"style":1497},[410],[394,3021,3023,3029,3080],{"className":3022},[415],[394,3024,3026],{"className":3025,"style":1226},[539,1225],[394,3027,540],{"className":3028},[1230,1507],[394,3030,3032],{"className":3031},[909],[394,3033,3035,3072],{"className":3034},[427,913],[394,3036,3038,3069],{"className":3037},[431],[394,3039,3041,3055],{"className":3040,"style":1520},[435],[394,3042,3043,3046],{"style":1523},[394,3044],{"className":3045,"style":443},[442],[394,3047,3049],{"className":3048},[447,448,449,450],[394,3050,3052],{"className":3051},[415,450],[394,3053,2068],{"className":3054,"style":2067},[415,419,450],[394,3056,3057,3060],{"style":1538},[394,3058],{"className":3059,"style":443},[442],[394,3061,3063],{"className":3062},[447,448,449,450],[394,3064,3066],{"className":3065},[415,450],[394,3067,605],{"className":3068},[415,419,450],[394,3070,983],{"className":3071},[982],[394,3073,3075],{"className":3074},[431],[394,3076,3078],{"className":3077,"style":1560},[435],[394,3079],{},[394,3081,3083],{"className":3082,"style":1226},[562,1225],[394,3084,563],{"className":3085},[1230,1507],[394,3087],{"className":3088,"style":466},[465],[394,3090,674],{"className":3091},[470],[394,3093],{"className":3094,"style":466},[465],[394,3096,3098,3101,3179,3182,3185],{"className":3097},[406],[394,3099],{"className":3100,"style":2181},[410],[394,3102,3104,3110,3173],{"className":3103},[415],[394,3105,3107],{"className":3106,"style":1226},[539,1225],[394,3108,540],{"className":3109},[1230,1507],[394,3111,3113],{"className":3112},[909],[394,3114,3116,3165],{"className":3115},[427,913],[394,3117,3119,3162],{"className":3118},[431],[394,3120,3122,3142],{"className":3121,"style":2203},[435],[394,3123,3124,3127],{"style":1523},[394,3125],{"className":3126,"style":443},[442],[394,3128,3130],{"className":3129},[447,448,449,450],[394,3131,3133,3136,3139],{"className":3132},[415,450],[394,3134,2068],{"className":3135,"style":2067},[415,419,450],[394,3137,457],{"className":3138},[516,450],[394,3140,461],{"className":3141},[415,450],[394,3143,3144,3147],{"style":1538},[394,3145],{"className":3146,"style":443},[442],[394,3148,3150],{"className":3149},[447,448,449,450],[394,3151,3153,3156,3159],{"className":3152},[415,450],[394,3154,605],{"className":3155},[415,419,450],[394,3157,457],{"className":3158},[516,450],[394,3160,461],{"className":3161},[415,450],[394,3163,983],{"className":3164},[982],[394,3166,3168],{"className":3167},[431],[394,3169,3171],{"className":3170,"style":1776},[435],[394,3172],{},[394,3174,3176],{"className":3175,"style":1226},[562,1225],[394,3177,563],{"className":3178},[1230,1507],[394,3180],{"className":3181,"style":626},[465],[394,3183,1080],{"className":3184},[516],[394,3186],{"className":3187,"style":626},[465],[394,3189,3191,3194],{"className":3190},[406],[394,3192],{"className":3193,"style":2276},[410],[394,3195,3197,3203,3260],{"className":3196},[415],[394,3198,3200],{"className":3199,"style":1226},[539,1225],[394,3201,540],{"className":3202},[1230,1507],[394,3204,3206],{"className":3205},[909],[394,3207,3209,3252],{"className":3208},[427,913],[394,3210,3212,3249],{"className":3211},[431],[394,3213,3215,3229],{"className":3214,"style":2203},[435],[394,3216,3217,3220],{"style":1523},[394,3218],{"className":3219,"style":443},[442],[394,3221,3223],{"className":3222},[447,448,449,450],[394,3224,3226],{"className":3225},[415,450],[394,3227,2068],{"className":3228,"style":2067},[415,419,450],[394,3230,3231,3234],{"style":1538},[394,3232],{"className":3233,"style":443},[442],[394,3235,3237],{"className":3236},[447,448,449,450],[394,3238,3240,3243,3246],{"className":3239},[415,450],[394,3241,605],{"className":3242},[415,419,450],[394,3244,457],{"className":3245},[516,450],[394,3247,461],{"className":3248},[415,450],[394,3250,983],{"className":3251},[982],[394,3253,3255],{"className":3254},[431],[394,3256,3258],{"className":3257,"style":1560},[435],[394,3259],{},[394,3261,3263],{"className":3262,"style":1226},[562,1225],[394,3264,563],{"className":3265},[1230,1507],[381,3267,3268,3269,3567,3568,3583,3584,392,3636,3639,3640,3655,3656,3659,3660,3663,3664,3768],{},"The highlighted cell is ",[394,3270,3272],{"className":3271},[397],[394,3273,3275,3357,3376,3396,3414,3495],{"className":3274,"ariaHidden":402},[401],[394,3276,3278,3281,3348,3351,3354],{"className":3277},[406],[394,3279],{"className":3280,"style":2276},[410],[394,3282,3284,3290,3342],{"className":3283},[415],[394,3285,3287],{"className":3286,"style":1226},[539,1225],[394,3288,540],{"className":3289},[1230,1507],[394,3291,3293],{"className":3292},[909],[394,3294,3296,3334],{"className":3295},[427,913],[394,3297,3299,3331],{"className":3298},[431],[394,3300,3302,3316],{"className":3301,"style":2203},[435],[394,3303,3304,3307],{"style":1523},[394,3305],{"className":3306,"style":443},[442],[394,3308,3310],{"className":3309},[447,448,449,450],[394,3311,3313],{"className":3312},[415,450],[394,3314,520],{"className":3315},[415,450],[394,3317,3318,3321],{"style":1538},[394,3319],{"className":3320,"style":443},[442],[394,3322,3324],{"className":3323},[447,448,449,450],[394,3325,3327],{"className":3326},[415,450],[394,3328,3330],{"className":3329},[415,450],"4",[394,3332,983],{"className":3333},[982],[394,3335,3337],{"className":3336},[431],[394,3338,3340],{"className":3339,"style":1560},[435],[394,3341],{},[394,3343,3345],{"className":3344,"style":1226},[562,1225],[394,3346,563],{"className":3347},[1230,1507],[394,3349],{"className":3350,"style":466},[465],[394,3352,674],{"className":3353},[470],[394,3355],{"className":3356,"style":466},[465],[394,3358,3360,3363,3367,3370,3373],{"className":3359},[406],[394,3361],{"className":3362,"style":639},[410],[394,3364,3366],{"className":3365},[415],"6",[394,3368],{"className":3369,"style":466},[465],[394,3371,674],{"className":3372},[470],[394,3374],{"className":3375,"style":466},[465],[394,3377,3379,3383,3387,3390,3393],{"className":3378},[406],[394,3380],{"className":3381,"style":3382},[410],"height:0.7278em;vertical-align:-0.0833em;",[394,3384,3386],{"className":3385},[415],"3",[394,3388],{"className":3389,"style":626},[465],[394,3391,1080],{"className":3392},[516],[394,3394],{"className":3395,"style":626},[465],[394,3397,3399,3402,3405,3408,3411],{"className":3398},[406],[394,3400],{"className":3401,"style":639},[410],[394,3403,3386],{"className":3404},[415],[394,3406],{"className":3407,"style":466},[465],[394,3409,674],{"className":3410},[470],[394,3412],{"className":3413,"style":466},[465],[394,3415,3417,3420,3486,3489,3492],{"className":3416},[406],[394,3418],{"className":3419,"style":2276},[410],[394,3421,3423,3429,3480],{"className":3422},[415],[394,3424,3426],{"className":3425,"style":1226},[539,1225],[394,3427,540],{"className":3428},[1230,1507],[394,3430,3432],{"className":3431},[909],[394,3433,3435,3472],{"className":3434},[427,913],[394,3436,3438,3469],{"className":3437},[431],[394,3439,3441,3455],{"className":3440,"style":2203},[435],[394,3442,3443,3446],{"style":1523},[394,3444],{"className":3445,"style":443},[442],[394,3447,3449],{"className":3448},[447,448,449,450],[394,3450,3452],{"className":3451},[415,450],[394,3453,461],{"className":3454},[415,450],[394,3456,3457,3460],{"style":1538},[394,3458],{"className":3459,"style":443},[442],[394,3461,3463],{"className":3462},[447,448,449,450],[394,3464,3466],{"className":3465},[415,450],[394,3467,3386],{"className":3468},[415,450],[394,3470,983],{"className":3471},[982],[394,3473,3475],{"className":3474},[431],[394,3476,3478],{"className":3477,"style":1560},[435],[394,3479],{},[394,3481,3483],{"className":3482,"style":1226},[562,1225],[394,3484,563],{"className":3485},[1230,1507],[394,3487],{"className":3488,"style":626},[465],[394,3490,1080],{"className":3491},[516],[394,3493],{"className":3494,"style":626},[465],[394,3496,3498,3501],{"className":3497},[406],[394,3499],{"className":3500,"style":2276},[410],[394,3502,3504,3510,3561],{"className":3503},[415],[394,3505,3507],{"className":3506,"style":1226},[539,1225],[394,3508,540],{"className":3509},[1230,1507],[394,3511,3513],{"className":3512},[909],[394,3514,3516,3553],{"className":3515},[427,913],[394,3517,3519,3550],{"className":3518},[431],[394,3520,3522,3536],{"className":3521,"style":2203},[435],[394,3523,3524,3527],{"style":1523},[394,3525],{"className":3526,"style":443},[442],[394,3528,3530],{"className":3529},[447,448,449,450],[394,3531,3533],{"className":3532},[415,450],[394,3534,520],{"className":3535},[415,450],[394,3537,3538,3541],{"style":1538},[394,3539],{"className":3540,"style":443},[442],[394,3542,3544],{"className":3543},[447,448,449,450],[394,3545,3547],{"className":3546},[415,450],[394,3548,3386],{"className":3549},[415,450],[394,3551,983],{"className":3552},[982],[394,3554,3556],{"className":3555},[431],[394,3557,3559],{"className":3558,"style":1560},[435],[394,3560],{},[394,3562,3564],{"className":3563,"style":1226},[562,1225],[394,3565,563],{"className":3566},[1230,1507],".\nBecause each entry needs only the row above, the whole triangle up to row ",[394,3569,3571],{"className":3570},[397],[394,3572,3574],{"className":3573,"ariaHidden":402},[401],[394,3575,3577,3580],{"className":3576},[406],[394,3578],{"className":3579,"style":601},[410],[394,3581,605],{"className":3582},[415,419]," is an\n",[394,3585,3587],{"className":3586},[397],[394,3588,3590],{"className":3589,"ariaHidden":402},[401],[394,3591,3593,3597,3601,3604,3633],{"className":3592},[406],[394,3594],{"className":3595,"style":3596},[410],"height:1.0641em;vertical-align:-0.25em;",[394,3598,3600],{"className":3599,"style":821},[415,419],"O",[394,3602,540],{"className":3603},[539],[394,3605,3607,3610],{"className":3606},[415],[394,3608,605],{"className":3609},[415,419],[394,3611,3613],{"className":3612},[423],[394,3614,3616],{"className":3615},[427],[394,3617,3619],{"className":3618},[431],[394,3620,3622],{"className":3621,"style":411},[435],[394,3623,3624,3627],{"style":438},[394,3625],{"className":3626,"style":443},[442],[394,3628,3630],{"className":3629},[447,448,449,450],[394,3631,520],{"className":3632},[415,450],[394,3634,563],{"className":3635},[562],[384,3637,3638],{"href":236},"dynamic program",", the right approach when ",[394,3641,3643],{"className":3642},[397],[394,3644,3646],{"className":3645,"ariaHidden":402},[401],[394,3647,3649,3652],{"className":3648},[406],[394,3650],{"className":3651,"style":601},[410],[394,3653,605],{"className":3654},[415,419]," is small or when no modulus is\ninvolved (and the basis for ",[1107,3657,3658],{},"Pascal's Triangle II"," and grid-path problems like\n",[1107,3661,3662],{},"Unique Paths",", whose answer is exactly ",[394,3665,3667],{"className":3666},[397],[394,3668,3670],{"className":3669,"ariaHidden":402},[401],[394,3671,3673,3676],{"className":3672},[406],[394,3674],{"className":3675,"style":2181},[410],[394,3677,3679,3685,3762],{"className":3678},[415],[394,3680,3682],{"className":3681,"style":1226},[539,1225],[394,3683,540],{"className":3684},[1230,1507],[394,3686,3688],{"className":3687},[909],[394,3689,3691,3754],{"className":3690},[427,913],[394,3692,3694,3751],{"className":3693},[431],[394,3695,3697,3725],{"className":3696,"style":2203},[435],[394,3698,3699,3702],{"style":1523},[394,3700],{"className":3701,"style":443},[442],[394,3703,3705],{"className":3704},[447,448,449,450],[394,3706,3708,3712,3716,3719,3722],{"className":3707},[415,450],[394,3709],{"className":3710,"style":3711},[465,450],"margin-right:0.1952em;",[394,3713,3715],{"className":3714},[415,419,450],"m",[394,3717,457],{"className":3718},[516,450],[394,3720,461],{"className":3721},[415,450],[394,3723],{"className":3724,"style":3711},[465,450],[394,3726,3727,3730],{"style":1538},[394,3728],{"className":3729,"style":443},[442],[394,3731,3733],{"className":3732},[447,448,449,450],[394,3734,3736,3739,3742,3745,3748],{"className":3735},[415,450],[394,3737,3715],{"className":3738},[415,419,450],[394,3740,1080],{"className":3741},[516,450],[394,3743,605],{"className":3744},[415,419,450],[394,3746,457],{"className":3747},[516,450],[394,3749,520],{"className":3750},[415,450],[394,3752,983],{"className":3753},[982],[394,3755,3757],{"className":3756},[431],[394,3758,3760],{"className":3759,"style":1776},[435],[394,3761],{},[394,3763,3765],{"className":3764,"style":1226},[562,1225],[394,3766,563],{"className":3767},[1230,1507],").",[381,3770,3771,3772,3806,3807,1099],{},"That grid-path count is Pascal's rule in disguise: label each lattice node with the\nnumber of monotone (right\u002Fdown) paths reaching it, and each node is the sum of its\nleft and top neighbours, exactly the additive recurrence. On a ",[394,3773,3775],{"className":3774},[397],[394,3776,3778,3797],{"className":3777,"ariaHidden":402},[401],[394,3779,3781,3784,3787,3790,3794],{"className":3780},[406],[394,3782],{"className":3783,"style":3382},[410],[394,3785,3386],{"className":3786},[415],[394,3788],{"className":3789,"style":626},[465],[394,3791,3793],{"className":3792},[516],"×",[394,3795],{"className":3796,"style":626},[465],[394,3798,3800,3803],{"className":3799},[406],[394,3801],{"className":3802,"style":639},[410],[394,3804,3386],{"className":3805},[415]," grid of\nnodes the corner reads ",[394,3808,3810],{"className":3809},[397],[394,3811,3813,3894],{"className":3812,"ariaHidden":402},[401],[394,3814,3816,3819,3885,3888,3891],{"className":3815},[406],[394,3817],{"className":3818,"style":2276},[410],[394,3820,3822,3828,3879],{"className":3821},[415],[394,3823,3825],{"className":3824,"style":1226},[539,1225],[394,3826,540],{"className":3827},[1230,1507],[394,3829,3831],{"className":3830},[909],[394,3832,3834,3871],{"className":3833},[427,913],[394,3835,3837,3868],{"className":3836},[431],[394,3838,3840,3854],{"className":3839,"style":2203},[435],[394,3841,3842,3845],{"style":1523},[394,3843],{"className":3844,"style":443},[442],[394,3846,3848],{"className":3847},[447,448,449,450],[394,3849,3851],{"className":3850},[415,450],[394,3852,520],{"className":3853},[415,450],[394,3855,3856,3859],{"style":1538},[394,3857],{"className":3858,"style":443},[442],[394,3860,3862],{"className":3861},[447,448,449,450],[394,3863,3865],{"className":3864},[415,450],[394,3866,3330],{"className":3867},[415,450],[394,3869,983],{"className":3870},[982],[394,3872,3874],{"className":3873},[431],[394,3875,3877],{"className":3876,"style":1560},[435],[394,3878],{},[394,3880,3882],{"className":3881,"style":1226},[562,1225],[394,3883,563],{"className":3884},[1230,1507],[394,3886],{"className":3887,"style":466},[465],[394,3889,674],{"className":3890},[470],[394,3892],{"className":3893,"style":466},[465],[394,3895,3897,3900],{"className":3896},[406],[394,3898],{"className":3899,"style":639},[410],[394,3901,3366],{"className":3902},[415],[2817,3904,3906,4081],{"className":3905},[2820,2821],[2823,3907,3911],{"xmlns":2825,"width":3908,"height":3909,"viewBox":3910},"210.091","166.757","-75 -75 157.568 125.068",[2830,3912,3913,3917,3928,3939,3950,3961,3973,3985,3996,4007,4019],{"stroke":2832,"style":2833},[2835,3914],{"fill":2837,"stroke":3915,"d":3916},"var(--tk-soft-neutral)","M-21.605-62.112h17.071M-32.987-50.73v17.071M18.228-62.112H35.3M-32.987-10.897V6.175M-21.605-22.278h17.071M6.847-50.73v17.071M18.228-22.278H35.3M6.847-10.897V6.175M-21.605 17.556h17.071M46.681-50.73v17.071M18.228 17.556H35.3M46.681-10.897V6.175",[2830,3918,3919,3922],{"fill":3915},[2835,3920],{"d":3921},"M-23.028-62.112c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3923,3924],{"transform":2841},[2835,3925],{"d":3926,"fill":2832,"stroke":2832,"className":3927,"style":2847},"M-29.080-62.112L-32.112-62.112L-32.112-62.428Q-30.961-62.428-30.961-62.723L-30.961-67.447Q-31.449-67.214-32.170-67.214L-32.170-67.530Q-31.040-67.530-30.478-68.106L-30.333-68.106Q-30.298-68.106-30.265-68.073Q-30.232-68.040-30.232-68.005L-30.232-62.723Q-30.232-62.428-29.080-62.428",[2846],[2830,3929,3930,3933],{"fill":3915},[2835,3931],{"d":3932},"M16.806-62.112c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3934,3936],{"transform":3935},"translate(37.521 2.9)",[2835,3937],{"d":3926,"fill":2832,"stroke":2832,"className":3938,"style":2847},[2846],[2830,3940,3941,3944],{"fill":3915},[2835,3942],{"d":3943},"M56.64-62.112c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3945,3947],{"transform":3946},"translate(77.355 2.9)",[2835,3948],{"d":3926,"fill":2832,"stroke":2832,"className":3949,"style":2847},[2846],[2830,3951,3952,3955],{"fill":3915},[2835,3953],{"d":3954},"M-23.028-22.278c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3956,3958],{"transform":3957},"translate(-2.312 42.734)",[2835,3959],{"d":3926,"fill":2832,"stroke":2832,"className":3960,"style":2847},[2846],[2830,3962,3963,3966],{"fill":3915},[2835,3964],{"d":3965},"M16.806-22.278c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3967,3969],{"transform":3968},"translate(37.521 42.734)",[2835,3970],{"d":3971,"fill":2832,"stroke":2832,"className":3972,"style":2847},"M-29.080-62.112L-32.530-62.112L-32.530-62.345Q-32.530-62.358-32.499-62.389L-31.045-63.966Q-30.579-64.463-30.326-64.768Q-30.073-65.074-29.882-65.485Q-29.691-65.896-29.691-66.335Q-29.691-66.924-30.014-67.357Q-30.337-67.790-30.917-67.790Q-31.181-67.790-31.427-67.680Q-31.673-67.570-31.849-67.383Q-32.025-67.196-32.121-66.946L-32.042-66.946Q-31.840-66.946-31.697-66.810Q-31.554-66.674-31.554-66.458Q-31.554-66.252-31.697-66.113Q-31.840-65.975-32.042-65.975Q-32.244-65.975-32.387-66.118Q-32.530-66.260-32.530-66.458Q-32.530-66.920-32.293-67.293Q-32.055-67.667-31.655-67.886Q-31.256-68.106-30.807-68.106Q-30.284-68.106-29.830-67.891Q-29.375-67.675-29.102-67.276Q-28.830-66.876-28.830-66.335Q-28.830-65.940-29.001-65.586Q-29.173-65.232-29.438-64.953Q-29.704-64.674-30.155-64.289Q-30.605-63.905-30.684-63.830L-31.708-62.868L-30.891-62.868Q-30.240-62.868-29.803-62.879Q-29.366-62.890-29.335-62.912Q-29.265-62.995-29.210-63.235Q-29.155-63.474-29.115-63.742L-28.830-63.742",[2846],[2830,3974,3975,3978],{"fill":3915},[2835,3976],{"d":3977},"M56.64-22.278c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3979,3981],{"transform":3980},"translate(77.355 42.734)",[2835,3982],{"d":3983,"fill":2832,"stroke":2832,"className":3984,"style":2847},"M-32.086-62.833L-32.130-62.833Q-31.928-62.516-31.541-62.358Q-31.154-62.200-30.728-62.200Q-30.192-62.200-29.953-62.635Q-29.713-63.070-29.713-63.650Q-29.713-64.230-29.959-64.670Q-30.205-65.109-30.737-65.109L-31.357-65.109Q-31.383-65.109-31.416-65.138Q-31.449-65.166-31.449-65.188L-31.449-65.289Q-31.449-65.320-31.420-65.344Q-31.392-65.368-31.357-65.368L-30.838-65.408Q-30.372-65.408-30.126-65.880Q-29.880-66.353-29.880-66.871Q-29.880-67.298-30.093-67.572Q-30.306-67.847-30.728-67.847Q-31.071-67.847-31.396-67.717Q-31.721-67.588-31.906-67.333L-31.880-67.333Q-31.677-67.333-31.541-67.192Q-31.405-67.051-31.405-66.854Q-31.405-66.656-31.539-66.522Q-31.673-66.388-31.871-66.388Q-32.073-66.388-32.211-66.522Q-32.350-66.656-32.350-66.854Q-32.350-67.443-31.847-67.774Q-31.343-68.106-30.728-68.106Q-30.350-68.106-29.948-67.966Q-29.546-67.825-29.278-67.546Q-29.010-67.267-29.010-66.871Q-29.010-66.322-29.364-65.885Q-29.717-65.447-30.258-65.263Q-29.867-65.184-29.522-64.960Q-29.177-64.736-28.966-64.395Q-28.755-64.054-28.755-63.659Q-28.755-63.277-28.918-62.954Q-29.080-62.631-29.372-62.395Q-29.665-62.160-30.012-62.037Q-30.359-61.914-30.728-61.914Q-31.176-61.914-31.607-62.075Q-32.038-62.235-32.319-62.562Q-32.600-62.890-32.600-63.347Q-32.600-63.562-32.453-63.705Q-32.306-63.848-32.086-63.848Q-31.875-63.848-31.730-63.703Q-31.585-63.558-31.585-63.347Q-31.585-63.136-31.732-62.984Q-31.880-62.833-32.086-62.833",[2846],[2830,3986,3987,3990],{"fill":3915},[2835,3988],{"d":3989},"M-23.028 17.556c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,3991,3993],{"transform":3992},"translate(-2.312 82.568)",[2835,3994],{"d":3926,"fill":2832,"stroke":2832,"className":3995,"style":2847},[2846],[2830,3997,3998,4001],{"fill":3915},[2835,3999],{"d":4000},"M16.806 17.556c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[2830,4002,4004],{"transform":4003},"translate(37.521 82.568)",[2835,4005],{"d":3983,"fill":2832,"stroke":2832,"className":4006,"style":2847},[2846],[2830,4008,4009,4012],{"fill":2952,"stroke":2953,"style":2954},[2835,4010],{"d":4011},"M58.062 17.556c0-6.286-5.095-11.381-11.38-11.381-6.287 0-11.382 5.095-11.382 11.381s5.095 11.381 11.381 11.381 11.381-5.095 11.381-11.38Zm-11.38 0",[2830,4013,4015],{"transform":4014},"translate(77.355 82.568)",[2835,4016],{"d":4017,"fill":2832,"stroke":2832,"className":4018,"style":2847},"M-30.675-61.914Q-31.409-61.914-31.840-62.395Q-32.271-62.877-32.435-63.569Q-32.600-64.261-32.600-65.008Q-32.600-65.737-32.308-66.460Q-32.016-67.183-31.462-67.645Q-30.908-68.106-30.161-68.106Q-29.665-68.106-29.329-67.840Q-28.992-67.574-28.992-67.091Q-28.992-66.911-29.120-66.783Q-29.247-66.656-29.423-66.656Q-29.603-66.656-29.733-66.781Q-29.862-66.906-29.862-67.091Q-29.862-67.205-29.805-67.309Q-29.748-67.412-29.647-67.471Q-29.546-67.530-29.423-67.530Q-29.419-67.530-29.414-67.528Q-29.410-67.526-29.405-67.522Q-29.520-67.689-29.728-67.768Q-29.937-67.847-30.161-67.847Q-30.605-67.847-30.963-67.546Q-31.321-67.245-31.510-66.792Q-31.743-66.186-31.743-65.153Q-31.572-65.518-31.271-65.746Q-30.970-65.975-30.583-65.975Q-30.179-65.975-29.834-65.808Q-29.489-65.641-29.252-65.360Q-29.014-65.078-28.885-64.716Q-28.755-64.353-28.755-63.949Q-28.755-63.404-28.999-62.938Q-29.243-62.472-29.682-62.193Q-30.122-61.914-30.675-61.914M-30.675-62.200Q-30.214-62.200-29.979-62.457Q-29.744-62.714-29.678-63.088Q-29.612-63.461-29.612-63.931L-29.612-63.966Q-29.612-64.454-29.669-64.819Q-29.726-65.184-29.955-65.447Q-30.183-65.711-30.627-65.711Q-30.996-65.711-31.247-65.467Q-31.497-65.223-31.612-64.859Q-31.726-64.494-31.726-64.147Q-31.726-64.028-31.717-63.966Q-31.717-63.949-31.719-63.938Q-31.721-63.927-31.726-63.914Q-31.726-63.263-31.488-62.732Q-31.251-62.200-30.675-62.200",[2846],[2830,4020,4021],{"fill":2953,"stroke":2953},[2830,4022,4025,4033,4039,4045,4051,4057,4063,4069,4075],{"fill":2953,"stroke":2837,"fontFamily":4023,"fontSize":4024},"cmr8","8",[2830,4026,4028],{"transform":4027},"translate(-29.284 103.576)",[2835,4029],{"d":4030,"fill":2953,"stroke":2953,"className":4031,"style":4032},"M-32.706-62.120L-32.706-63.342Q-32.706-63.370-32.675-63.401Q-32.643-63.432-32.620-63.432L-32.514-63.432Q-32.444-63.432-32.428-63.370Q-32.366-63.050-32.227-62.809Q-32.089-62.569-31.856-62.428Q-31.624-62.288-31.315-62.288Q-31.077-62.288-30.868-62.348Q-30.659-62.409-30.522-62.557Q-30.385-62.706-30.385-62.952Q-30.385-63.206-30.596-63.372Q-30.807-63.538-31.077-63.592L-31.698-63.706Q-32.104-63.784-32.405-64.040Q-32.706-64.296-32.706-64.671Q-32.706-65.038-32.505-65.260Q-32.303-65.483-31.979-65.581Q-31.655-65.678-31.315-65.678Q-30.850-65.678-30.553-65.471L-30.331-65.655Q-30.307-65.678-30.276-65.678L-30.225-65.678Q-30.194-65.678-30.167-65.651Q-30.139-65.624-30.139-65.592L-30.139-64.608Q-30.139-64.577-30.165-64.548Q-30.190-64.518-30.225-64.518L-30.331-64.518Q-30.366-64.518-30.393-64.546Q-30.421-64.573-30.421-64.608Q-30.421-65.007-30.673-65.227Q-30.925-65.448-31.323-65.448Q-31.678-65.448-31.962-65.325Q-32.245-65.202-32.245-64.897Q-32.245-64.678-32.044-64.546Q-31.842-64.413-31.596-64.370L-30.971-64.257Q-30.542-64.167-30.233-63.870Q-29.925-63.573-29.925-63.159Q-29.925-62.589-30.323-62.311Q-30.721-62.034-31.315-62.034Q-31.866-62.034-32.217-62.370L-32.514-62.057Q-32.538-62.034-32.573-62.034L-32.620-62.034Q-32.643-62.034-32.675-62.065Q-32.706-62.096-32.706-62.120M-28.772-63.073L-28.772-65.264L-29.475-65.264L-29.475-65.518Q-29.120-65.518-28.878-65.751Q-28.635-65.983-28.524-66.331Q-28.413-66.678-28.413-67.034L-28.132-67.034L-28.132-65.561L-26.956-65.561L-26.956-65.264L-28.132-65.264L-28.132-63.089Q-28.132-62.768-28.012-62.540Q-27.893-62.311-27.612-62.311Q-27.432-62.311-27.315-62.434Q-27.198-62.557-27.145-62.737Q-27.092-62.917-27.092-63.089L-27.092-63.561L-26.811-63.561L-26.811-63.073Q-26.811-62.819-26.917-62.579Q-27.022-62.339-27.219-62.186Q-27.417-62.034-27.675-62.034Q-27.991-62.034-28.243-62.157Q-28.495-62.280-28.633-62.514Q-28.772-62.749-28.772-63.073M-25.995-62.944Q-25.995-63.428-25.592-63.723Q-25.190-64.018-24.639-64.137Q-24.089-64.257-23.596-64.257L-23.596-64.546Q-23.596-64.772-23.712-64.979Q-23.827-65.186-24.024-65.305Q-24.221-65.424-24.452-65.424Q-24.878-65.424-25.163-65.319Q-25.092-65.292-25.046-65.237Q-24.999-65.182-24.973-65.112Q-24.948-65.042-24.948-64.967Q-24.948-64.862-24.999-64.770Q-25.050-64.678-25.141-64.628Q-25.233-64.577-25.339-64.577Q-25.444-64.577-25.536-64.628Q-25.628-64.678-25.678-64.770Q-25.729-64.862-25.729-64.967Q-25.729-65.385-25.341-65.532Q-24.952-65.678-24.452-65.678Q-24.120-65.678-23.766-65.548Q-23.413-65.417-23.184-65.163Q-22.956-64.909-22.956-64.561L-22.956-62.760Q-22.956-62.628-22.883-62.518Q-22.811-62.409-22.682-62.409Q-22.557-62.409-22.489-62.514Q-22.421-62.620-22.421-62.760L-22.421-63.272L-22.139-63.272L-22.139-62.760Q-22.139-62.557-22.257-62.399Q-22.374-62.241-22.555-62.157Q-22.737-62.073-22.940-62.073Q-23.171-62.073-23.323-62.245Q-23.475-62.417-23.507-62.647Q-23.667-62.366-23.975-62.200Q-24.284-62.034-24.635-62.034Q-25.147-62.034-25.571-62.257Q-25.995-62.479-25.995-62.944M-25.307-62.944Q-25.307-62.659-25.081-62.473Q-24.854-62.288-24.561-62.288Q-24.315-62.288-24.091-62.405Q-23.866-62.522-23.731-62.725Q-23.596-62.928-23.596-63.182L-23.596-64.014Q-23.862-64.014-24.147-63.960Q-24.432-63.905-24.704-63.776Q-24.975-63.647-25.141-63.440Q-25.307-63.233-25.307-62.944M-19.839-62.112L-21.819-62.112L-21.819-62.409Q-21.550-62.409-21.382-62.454Q-21.214-62.499-21.214-62.671L-21.214-64.807Q-21.214-65.022-21.276-65.118Q-21.339-65.214-21.456-65.235Q-21.573-65.257-21.819-65.257L-21.819-65.553L-20.651-65.639L-20.651-64.854Q-20.573-65.065-20.421-65.251Q-20.268-65.436-20.069-65.538Q-19.870-65.639-19.643-65.639Q-19.397-65.639-19.206-65.495Q-19.014-65.350-19.014-65.120Q-19.014-64.964-19.120-64.854Q-19.225-64.745-19.382-64.745Q-19.538-64.745-19.647-64.854Q-19.757-64.964-19.757-65.120Q-19.757-65.280-19.651-65.385Q-19.975-65.385-20.190-65.157Q-20.405-64.928-20.501-64.589Q-20.596-64.249-20.596-63.944L-20.596-62.671Q-20.596-62.503-20.370-62.456Q-20.143-62.409-19.839-62.409L-19.839-62.112M-17.909-63.073L-17.909-65.264L-18.612-65.264L-18.612-65.518Q-18.257-65.518-18.014-65.751Q-17.772-65.983-17.661-66.331Q-17.550-66.678-17.550-67.034L-17.268-67.034L-17.268-65.561L-16.092-65.561L-16.092-65.264L-17.268-65.264L-17.268-63.089Q-17.268-62.768-17.149-62.540Q-17.030-62.311-16.749-62.311Q-16.569-62.311-16.452-62.434Q-16.335-62.557-16.282-62.737Q-16.229-62.917-16.229-63.089L-16.229-63.561L-15.948-63.561L-15.948-63.073Q-15.948-62.819-16.053-62.579Q-16.159-62.339-16.356-62.186Q-16.553-62.034-16.811-62.034Q-17.128-62.034-17.380-62.157Q-17.632-62.280-17.770-62.514Q-17.909-62.749-17.909-63.073",[2846],"stroke-width:0.240",[2830,4034,4035],{"transform":4027},[2835,4036],{"d":4037,"fill":2953,"stroke":2953,"className":4038,"style":4032},"M-11.764-63.073L-11.764-65.264L-12.467-65.264L-12.467-65.518Q-12.111-65.518-11.869-65.751Q-11.627-65.983-11.516-66.331Q-11.404-66.678-11.404-67.034L-11.123-67.034L-11.123-65.561L-9.947-65.561L-9.947-65.264L-11.123-65.264L-11.123-63.089Q-11.123-62.768-11.004-62.540Q-10.885-62.311-10.604-62.311Q-10.424-62.311-10.307-62.434Q-10.190-62.557-10.137-62.737Q-10.084-62.917-10.084-63.089L-10.084-63.561L-9.803-63.561L-9.803-63.073Q-9.803-62.819-9.908-62.579Q-10.014-62.339-10.211-62.186Q-10.408-62.034-10.666-62.034Q-10.982-62.034-11.234-62.157Q-11.486-62.280-11.625-62.514Q-11.764-62.749-11.764-63.073M-9.084-63.807Q-9.084-64.311-8.828-64.743Q-8.572-65.174-8.137-65.426Q-7.701-65.678-7.201-65.678Q-6.815-65.678-6.473-65.534Q-6.131-65.389-5.869-65.128Q-5.607-64.866-5.465-64.530Q-5.322-64.194-5.322-63.807Q-5.322-63.315-5.586-62.905Q-5.850-62.495-6.279-62.264Q-6.709-62.034-7.201-62.034Q-7.693-62.034-8.127-62.266Q-8.561-62.499-8.822-62.907Q-9.084-63.315-9.084-63.807M-7.201-62.311Q-6.744-62.311-6.492-62.534Q-6.240-62.757-6.152-63.108Q-6.065-63.460-6.065-63.905Q-6.065-64.335-6.158-64.673Q-6.252-65.010-6.506-65.217Q-6.760-65.424-7.201-65.424Q-7.850-65.424-8.094-65.008Q-8.338-64.592-8.338-63.905Q-8.338-63.460-8.250-63.108Q-8.162-62.757-7.910-62.534Q-7.658-62.311-7.201-62.311M-2.955-60.561L-4.811-60.561L-4.811-60.854Q-4.541-60.854-4.373-60.899Q-4.205-60.944-4.205-61.120L-4.205-64.944Q-4.205-65.151-4.361-65.204Q-4.518-65.257-4.811-65.257L-4.811-65.553L-3.588-65.639L-3.588-65.174Q-3.357-65.397-3.043-65.518Q-2.729-65.639-2.389-65.639Q-1.916-65.639-1.512-65.393Q-1.107-65.147-0.875-64.731Q-0.643-64.315-0.643-63.839Q-0.643-63.464-0.791-63.135Q-0.940-62.807-1.209-62.555Q-1.479-62.303-1.822-62.169Q-2.166-62.034-2.525-62.034Q-2.815-62.034-3.086-62.155Q-3.357-62.276-3.565-62.487L-3.565-61.120Q-3.565-60.944-3.397-60.899Q-3.229-60.854-2.955-60.854L-2.955-60.561M-3.565-64.776L-3.565-62.936Q-3.412-62.647-3.150-62.467Q-2.889-62.288-2.580-62.288Q-2.295-62.288-2.072-62.426Q-1.850-62.565-1.697-62.796Q-1.545-63.026-1.467-63.298Q-1.389-63.569-1.389-63.839Q-1.389-64.171-1.514-64.528Q-1.639-64.885-1.887-65.122Q-2.135-65.358-2.482-65.358Q-2.807-65.358-3.102-65.202Q-3.397-65.046-3.565-64.776M1.994-63.561L-0.260-63.561L-0.260-64.112L1.994-64.112L1.994-63.561M4.627-62.112L2.795-62.112L2.795-62.409Q3.068-62.409 3.236-62.456Q3.404-62.503 3.404-62.671L3.404-66.831Q3.404-67.046 3.342-67.141Q3.279-67.237 3.160-67.258Q3.041-67.280 2.795-67.280L2.795-67.577L4.018-67.663L4.018-62.671Q4.018-62.503 4.185-62.456Q4.353-62.409 4.627-62.409L4.627-62.112M5.072-63.866Q5.072-64.346 5.305-64.762Q5.537-65.178 5.947-65.428Q6.357-65.678 6.834-65.678Q7.564-65.678 7.963-65.237Q8.361-64.796 8.361-64.065Q8.361-63.960 8.268-63.936L5.818-63.936L5.818-63.866Q5.818-63.456 5.939-63.100Q6.060-62.745 6.332-62.528Q6.603-62.311 7.033-62.311Q7.396-62.311 7.693-62.540Q7.990-62.768 8.092-63.120Q8.100-63.167 8.185-63.182L8.268-63.182Q8.361-63.155 8.361-63.073Q8.361-63.065 8.353-63.034Q8.291-62.807 8.152-62.624Q8.014-62.440 7.822-62.307Q7.631-62.175 7.412-62.104Q7.193-62.034 6.955-62.034Q6.584-62.034 6.246-62.171Q5.908-62.307 5.641-62.559Q5.373-62.811 5.223-63.151Q5.072-63.491 5.072-63.866M5.826-64.174L7.787-64.174Q7.787-64.479 7.685-64.770Q7.584-65.061 7.367-65.243Q7.150-65.424 6.834-65.424Q6.533-65.424 6.303-65.237Q6.072-65.049 5.949-64.758Q5.826-64.467 5.826-64.174M10.916-62.112L8.932-62.112L8.932-62.409Q9.205-62.409 9.373-62.456Q9.541-62.503 9.541-62.671L9.541-65.264L8.900-65.264L8.900-65.561L9.541-65.561L9.541-66.495Q9.541-66.760 9.658-66.997Q9.775-67.233 9.969-67.397Q10.162-67.561 10.410-67.653Q10.658-67.745 10.924-67.745Q11.209-67.745 11.434-67.587Q11.658-67.428 11.658-67.151Q11.658-66.995 11.553-66.885Q11.447-66.776 11.283-66.776Q11.127-66.776 11.018-66.885Q10.908-66.995 10.908-67.151Q10.908-67.358 11.068-67.464Q10.971-67.487 10.877-67.487Q10.646-67.487 10.475-67.331Q10.303-67.174 10.217-66.938Q10.131-66.702 10.131-66.479L10.131-65.561L11.100-65.561L11.100-65.264L10.154-65.264L10.154-62.671Q10.154-62.503 10.381-62.456Q10.607-62.409 10.916-62.409L10.916-62.112M12.068-63.073L12.068-65.264L11.365-65.264L11.365-65.518Q11.721-65.518 11.963-65.751Q12.205-65.983 12.316-66.331Q12.428-66.678 12.428-67.034L12.709-67.034L12.709-65.561L13.885-65.561L13.885-65.264L12.709-65.264L12.709-63.089Q12.709-62.768 12.828-62.540Q12.947-62.311 13.228-62.311Q13.408-62.311 13.525-62.434Q13.643-62.557 13.695-62.737Q13.748-62.917 13.748-63.089L13.748-63.561L14.029-63.561L14.029-63.073Q14.029-62.819 13.924-62.579Q13.818-62.339 13.621-62.186Q13.424-62.034 13.166-62.034Q12.850-62.034 12.598-62.157Q12.346-62.280 12.207-62.514Q12.068-62.749 12.068-63.073M15.334-60.706Q15.334-60.729 15.365-60.776Q15.658-61.038 15.824-61.405Q15.990-61.772 15.990-62.159L15.990-62.217Q15.861-62.112 15.693-62.112Q15.502-62.112 15.365-62.245Q15.228-62.378 15.228-62.577Q15.228-62.768 15.365-62.901Q15.502-63.034 15.693-63.034Q15.994-63.034 16.119-62.764Q16.244-62.495 16.244-62.159Q16.244-61.710 16.062-61.296Q15.881-60.882 15.541-60.585Q15.518-60.561 15.478-60.561Q15.432-60.561 15.383-60.606Q15.334-60.651 15.334-60.706",[2846],[2830,4040,4041],{"transform":4027},[2835,4042],{"d":4043,"fill":2953,"stroke":2953,"className":4044,"style":4032},"M19.959-63.807Q19.959-64.311 20.215-64.743Q20.471-65.174 20.907-65.426Q21.342-65.678 21.842-65.678Q22.229-65.678 22.571-65.534Q22.912-65.389 23.174-65.128Q23.436-64.866 23.578-64.530Q23.721-64.194 23.721-63.807Q23.721-63.315 23.457-62.905Q23.194-62.495 22.764-62.264Q22.334-62.034 21.842-62.034Q21.350-62.034 20.916-62.266Q20.483-62.499 20.221-62.907Q19.959-63.315 19.959-63.807M21.842-62.311Q22.299-62.311 22.551-62.534Q22.803-62.757 22.891-63.108Q22.979-63.460 22.979-63.905Q22.979-64.335 22.885-64.673Q22.791-65.010 22.537-65.217Q22.284-65.424 21.842-65.424Q21.194-65.424 20.950-65.008Q20.705-64.592 20.705-63.905Q20.705-63.460 20.793-63.108Q20.881-62.757 21.133-62.534Q21.385-62.311 21.842-62.311M26.135-62.112L24.280-62.112L24.280-62.409Q24.553-62.409 24.721-62.456Q24.889-62.503 24.889-62.671L24.889-64.807Q24.889-65.022 24.826-65.118Q24.764-65.214 24.645-65.235Q24.526-65.257 24.280-65.257L24.280-65.553L25.471-65.639L25.471-64.905Q25.584-65.120 25.778-65.288Q25.971-65.456 26.209-65.548Q26.448-65.639 26.701-65.639Q27.869-65.639 27.869-64.561L27.869-62.671Q27.869-62.503 28.039-62.456Q28.209-62.409 28.479-62.409L28.479-62.112L26.623-62.112L26.623-62.409Q26.897-62.409 27.065-62.456Q27.233-62.503 27.233-62.671L27.233-64.546Q27.233-64.928 27.112-65.157Q26.991-65.385 26.639-65.385Q26.326-65.385 26.073-65.223Q25.819-65.061 25.672-64.792Q25.526-64.522 25.526-64.225L25.526-62.671Q25.526-62.503 25.696-62.456Q25.866-62.409 26.135-62.409L26.135-62.112M30.838-62.112L29.006-62.112L29.006-62.409Q29.280-62.409 29.448-62.456Q29.616-62.503 29.616-62.671L29.616-66.831Q29.616-67.046 29.553-67.141Q29.491-67.237 29.371-67.258Q29.252-67.280 29.006-67.280L29.006-67.577L30.229-67.663L30.229-62.671Q30.229-62.503 30.397-62.456Q30.565-62.409 30.838-62.409L30.838-62.112M31.701-60.815Q31.815-60.737 31.991-60.737Q32.280-60.737 32.500-60.950Q32.721-61.163 32.846-61.464L33.135-62.112L31.862-64.999Q31.780-65.174 31.635-65.219Q31.491-65.264 31.221-65.264L31.221-65.561L32.940-65.561L32.940-65.264Q32.518-65.264 32.518-65.081Q32.518-65.069 32.534-64.999L33.471-62.874L34.303-64.784Q34.342-64.874 34.342-64.952Q34.342-65.092 34.241-65.178Q34.139-65.264 33.998-65.264L33.998-65.561L35.350-65.561L35.350-65.264Q35.096-65.264 34.903-65.139Q34.709-65.014 34.604-64.784L33.159-61.464Q33.045-61.210 32.879-60.987Q32.713-60.764 32.485-60.622Q32.256-60.479 31.991-60.479Q31.694-60.479 31.453-60.671Q31.213-60.862 31.213-61.151Q31.213-61.307 31.319-61.409Q31.424-61.510 31.573-61.510Q31.678-61.510 31.758-61.464Q31.838-61.417 31.885-61.339Q31.932-61.260 31.932-61.151Q31.932-61.030 31.871-60.942Q31.811-60.854 31.701-60.815",[2846],[2830,4046,4047],{"transform":4027},[2835,4048],{"d":4049,"fill":2953,"stroke":2953,"className":4050,"style":4032},"M40.620-62.112L38.640-62.112L38.640-62.409Q38.909-62.409 39.077-62.454Q39.245-62.499 39.245-62.671L39.245-64.807Q39.245-65.022 39.183-65.118Q39.120-65.214 39.003-65.235Q38.886-65.257 38.640-65.257L38.640-65.553L39.808-65.639L39.808-64.854Q39.886-65.065 40.038-65.251Q40.190-65.436 40.390-65.538Q40.589-65.639 40.815-65.639Q41.062-65.639 41.253-65.495Q41.444-65.350 41.444-65.120Q41.444-64.964 41.339-64.854Q41.233-64.745 41.077-64.745Q40.921-64.745 40.812-64.854Q40.702-64.964 40.702-65.120Q40.702-65.280 40.808-65.385Q40.483-65.385 40.269-65.157Q40.054-64.928 39.958-64.589Q39.862-64.249 39.862-63.944L39.862-62.671Q39.862-62.503 40.089-62.456Q40.315-62.409 40.620-62.409L40.620-62.112M43.784-62.112L42.007-62.112L42.007-62.409Q42.280-62.409 42.448-62.456Q42.616-62.503 42.616-62.671L42.616-64.807Q42.616-65.022 42.560-65.118Q42.503-65.214 42.390-65.235Q42.276-65.257 42.030-65.257L42.030-65.553L43.229-65.639L43.229-62.671Q43.229-62.503 43.376-62.456Q43.522-62.409 43.784-62.409L43.784-62.112M42.343-67.034Q42.343-67.225 42.478-67.356Q42.612-67.487 42.808-67.487Q42.929-67.487 43.032-67.424Q43.136-67.362 43.198-67.258Q43.261-67.155 43.261-67.034Q43.261-66.839 43.130-66.704Q42.999-66.569 42.808-66.569Q42.608-66.569 42.476-66.702Q42.343-66.835 42.343-67.034M44.284-61.503Q44.284-61.784 44.495-61.995Q44.706-62.206 44.991-62.296Q44.835-62.421 44.757-62.610Q44.679-62.800 44.679-62.999Q44.679-63.354 44.909-63.647Q44.542-63.987 44.542-64.456Q44.542-64.807 44.745-65.077Q44.948-65.346 45.269-65.493Q45.589-65.639 45.933-65.639Q46.452-65.639 46.823-65.358Q47.187-65.729 47.733-65.729Q47.913-65.729 48.040-65.602Q48.167-65.475 48.167-65.296Q48.167-65.190 48.089-65.112Q48.011-65.034 47.901-65.034Q47.792-65.034 47.716-65.110Q47.640-65.186 47.640-65.296Q47.640-65.397 47.679-65.448Q47.687-65.456 47.690-65.462Q47.694-65.467 47.694-65.471Q47.319-65.471 46.999-65.217Q47.319-64.878 47.319-64.456Q47.319-64.186 47.202-63.969Q47.085-63.753 46.880-63.594Q46.675-63.436 46.433-63.354Q46.190-63.272 45.933-63.272Q45.714-63.272 45.501-63.331Q45.288-63.389 45.093-63.510Q44.999-63.370 44.999-63.190Q44.999-62.983 45.136-62.831Q45.272-62.678 45.479-62.678L46.175-62.678Q46.663-62.678 47.075-62.594Q47.487-62.510 47.767-62.253Q48.046-61.995 48.046-61.503Q48.046-61.139 47.726-60.907Q47.405-60.675 46.964-60.573Q46.522-60.471 46.167-60.471Q45.812-60.471 45.368-60.573Q44.925-60.675 44.604-60.907Q44.284-61.139 44.284-61.503M44.788-61.503Q44.788-61.307 44.933-61.159Q45.077-61.010 45.290-60.921Q45.503-60.831 45.743-60.784Q45.983-60.737 46.167-60.737Q46.409-60.737 46.739-60.815Q47.069-60.893 47.306-61.067Q47.542-61.241 47.542-61.503Q47.542-61.909 47.132-62.018Q46.722-62.128 46.159-62.128L45.479-62.128Q45.210-62.128 44.999-61.950Q44.788-61.772 44.788-61.503M45.933-63.538Q46.655-63.538 46.655-64.456Q46.655-65.378 45.933-65.378Q45.206-65.378 45.206-64.456Q45.206-63.538 45.933-63.538M50.460-62.112L48.604-62.112L48.604-62.409Q48.878-62.409 49.046-62.456Q49.214-62.503 49.214-62.671L49.214-66.831Q49.214-67.046 49.151-67.141Q49.089-67.237 48.970-67.258Q48.851-67.280 48.604-67.280L48.604-67.577L49.827-67.663L49.827-64.960Q49.952-65.171 50.140-65.321Q50.327-65.471 50.554-65.555Q50.780-65.639 51.026-65.639Q52.194-65.639 52.194-64.561L52.194-62.671Q52.194-62.503 52.364-62.456Q52.534-62.409 52.804-62.409L52.804-62.112L50.948-62.112L50.948-62.409Q51.222-62.409 51.390-62.456Q51.558-62.503 51.558-62.671L51.558-64.546Q51.558-64.928 51.437-65.157Q51.315-65.385 50.964-65.385Q50.651-65.385 50.397-65.223Q50.144-65.061 49.997-64.792Q49.851-64.522 49.851-64.225L49.851-62.671Q49.851-62.503 50.020-62.456Q50.190-62.409 50.460-62.409",[2846],[2830,4052,4053],{"transform":4027},[2835,4054],{"d":4055,"fill":2953,"stroke":2953,"className":4056,"style":4032},"M53.647-63.073L53.647-65.264L52.944-65.264L52.944-65.518Q53.300-65.518 53.542-65.751Q53.784-65.983 53.895-66.331Q54.007-66.678 54.007-67.034L54.288-67.034L54.288-65.561L55.464-65.561L55.464-65.264L54.288-65.264L54.288-63.089Q54.288-62.768 54.407-62.540Q54.526-62.311 54.807-62.311Q54.987-62.311 55.104-62.434Q55.221-62.557 55.274-62.737Q55.327-62.917 55.327-63.089L55.327-63.561L55.608-63.561L55.608-63.073Q55.608-62.819 55.503-62.579Q55.397-62.339 55.200-62.186Q55.003-62.034 54.745-62.034Q54.429-62.034 54.177-62.157Q53.925-62.280 53.786-62.514Q53.647-62.749 53.647-63.073M56.561-60.296Q56.561-60.315 56.577-60.370L59.503-68.007Q59.569-68.112 59.675-68.112Q59.753-68.112 59.805-68.059Q59.858-68.007 59.858-67.928Q59.858-67.909 59.843-67.854L56.913-60.217Q56.850-60.112 56.745-60.112Q56.671-60.112 56.616-60.167Q56.561-60.221 56.561-60.296M62.389-62.034Q61.909-62.034 61.501-62.278Q61.093-62.522 60.854-62.936Q60.616-63.350 60.616-63.839Q60.616-64.331 60.874-64.747Q61.132-65.163 61.563-65.401Q61.995-65.639 62.487-65.639Q63.108-65.639 63.557-65.202L63.557-66.831Q63.557-67.046 63.495-67.141Q63.432-67.237 63.315-67.258Q63.198-67.280 62.952-67.280L62.952-67.577L64.175-67.663L64.175-62.854Q64.175-62.643 64.237-62.548Q64.300-62.452 64.417-62.430Q64.534-62.409 64.784-62.409L64.784-62.112L63.534-62.034L63.534-62.518Q63.069-62.034 62.389-62.034M62.456-62.288Q62.796-62.288 63.089-62.479Q63.382-62.671 63.534-62.967L63.534-64.799Q63.386-65.073 63.124-65.229Q62.862-65.385 62.550-65.385Q61.925-65.385 61.641-64.938Q61.358-64.491 61.358-63.831Q61.358-63.186 61.610-62.737Q61.862-62.288 62.456-62.288M65.292-63.807Q65.292-64.311 65.548-64.743Q65.804-65.174 66.239-65.426Q66.675-65.678 67.175-65.678Q67.561-65.678 67.903-65.534Q68.245-65.389 68.507-65.128Q68.768-64.866 68.911-64.530Q69.054-64.194 69.054-63.807Q69.054-63.315 68.790-62.905Q68.526-62.495 68.096-62.264Q67.667-62.034 67.175-62.034Q66.682-62.034 66.249-62.266Q65.815-62.499 65.554-62.907Q65.292-63.315 65.292-63.807M67.175-62.311Q67.632-62.311 67.884-62.534Q68.136-62.757 68.223-63.108Q68.311-63.460 68.311-63.905Q68.311-64.335 68.218-64.673Q68.124-65.010 67.870-65.217Q67.616-65.424 67.175-65.424Q66.526-65.424 66.282-65.008Q66.038-64.592 66.038-63.905Q66.038-63.460 66.126-63.108Q66.214-62.757 66.466-62.534Q66.718-62.311 67.175-62.311",[2846],[2830,4058,4059],{"transform":4027},[2835,4060],{"d":4061,"fill":2953,"stroke":2953,"className":4062,"style":4032},"M70.900-62.143L69.830-64.999Q69.763-65.178 69.633-65.221Q69.502-65.264 69.244-65.264L69.244-65.561L70.924-65.561L70.924-65.264Q70.474-65.264 70.474-65.065Q70.478-65.049 70.480-65.032Q70.482-65.014 70.482-64.999L71.275-62.905L71.986-64.815Q71.951-64.909 71.951-64.954Q71.951-64.999 71.916-64.999Q71.849-65.178 71.719-65.221Q71.588-65.264 71.334-65.264L71.334-65.561L72.924-65.561L72.924-65.264Q72.474-65.264 72.474-65.065Q72.478-65.046 72.480-65.028Q72.482-65.010 72.482-64.999L73.314-62.784L74.068-64.784Q74.092-64.842 74.092-64.913Q74.092-65.073 73.955-65.169Q73.818-65.264 73.650-65.264L73.650-65.561L75.037-65.561L75.037-65.264Q74.803-65.264 74.625-65.137Q74.447-65.010 74.365-64.784L73.381-62.143Q73.326-62.034 73.213-62.034L73.154-62.034Q73.041-62.034 72.998-62.143L72.138-64.417L71.283-62.143Q71.244-62.034 71.123-62.034L71.068-62.034Q70.955-62.034 70.900-62.143M77.381-62.112L75.525-62.112L75.525-62.409Q75.799-62.409 75.967-62.456Q76.135-62.503 76.135-62.671L76.135-64.807Q76.135-65.022 76.072-65.118Q76.010-65.214 75.890-65.235Q75.771-65.257 75.525-65.257L75.525-65.553L76.717-65.639L76.717-64.905Q76.830-65.120 77.023-65.288Q77.217-65.456 77.455-65.548Q77.693-65.639 77.947-65.639Q79.115-65.639 79.115-64.561L79.115-62.671Q79.115-62.503 79.285-62.456Q79.455-62.409 79.724-62.409L79.724-62.112L77.869-62.112L77.869-62.409Q78.142-62.409 78.310-62.456Q78.478-62.503 78.478-62.671L78.478-64.546Q78.478-64.928 78.357-65.157Q78.236-65.385 77.885-65.385Q77.572-65.385 77.318-65.223Q77.064-65.061 76.918-64.792Q76.771-64.522 76.771-64.225L76.771-62.671Q76.771-62.503 76.941-62.456Q77.111-62.409 77.381-62.409",[2846],[2830,4064,4065],{"transform":4027},[2835,4066],{"d":4067,"fill":2953,"stroke":2953,"className":4068,"style":4032},"M84.939-62.112L83.084-62.112L83.084-62.409Q83.357-62.409 83.525-62.456Q83.693-62.503 83.693-62.671L83.693-64.807Q83.693-65.022 83.630-65.118Q83.568-65.214 83.449-65.235Q83.330-65.257 83.084-65.257L83.084-65.553L84.275-65.639L84.275-64.905Q84.388-65.120 84.582-65.288Q84.775-65.456 85.013-65.548Q85.251-65.639 85.505-65.639Q86.466-65.639 86.642-64.928Q86.826-65.257 87.154-65.448Q87.482-65.639 87.861-65.639Q89.037-65.639 89.037-64.561L89.037-62.671Q89.037-62.503 89.205-62.456Q89.373-62.409 89.642-62.409L89.642-62.112L87.787-62.112L87.787-62.409Q88.060-62.409 88.228-62.454Q88.396-62.499 88.396-62.671L88.396-64.546Q88.396-64.932 88.271-65.159Q88.146-65.385 87.794-65.385Q87.490-65.385 87.234-65.223Q86.978-65.061 86.830-64.792Q86.681-64.522 86.681-64.225L86.681-62.671Q86.681-62.503 86.851-62.456Q87.021-62.409 87.291-62.409L87.291-62.112L85.435-62.112L85.435-62.409Q85.709-62.409 85.876-62.456Q86.044-62.503 86.044-62.671L86.044-64.546Q86.044-64.932 85.919-65.159Q85.794-65.385 85.443-65.385Q85.138-65.385 84.882-65.223Q84.626-65.061 84.478-64.792Q84.330-64.522 84.330-64.225L84.330-62.671Q84.330-62.503 84.500-62.456Q84.669-62.409 84.939-62.409L84.939-62.112M90.087-63.807Q90.087-64.311 90.343-64.743Q90.599-65.174 91.035-65.426Q91.470-65.678 91.970-65.678Q92.357-65.678 92.699-65.534Q93.041-65.389 93.302-65.128Q93.564-64.866 93.707-64.530Q93.849-64.194 93.849-63.807Q93.849-63.315 93.585-62.905Q93.322-62.495 92.892-62.264Q92.462-62.034 91.970-62.034Q91.478-62.034 91.044-62.266Q90.611-62.499 90.349-62.907Q90.087-63.315 90.087-63.807M91.970-62.311Q92.427-62.311 92.679-62.534Q92.931-62.757 93.019-63.108Q93.107-63.460 93.107-63.905Q93.107-64.335 93.013-64.673Q92.919-65.010 92.666-65.217Q92.412-65.424 91.970-65.424Q91.322-65.424 91.078-65.008Q90.834-64.592 90.834-63.905Q90.834-63.460 90.921-63.108Q91.009-62.757 91.261-62.534Q91.513-62.311 91.970-62.311",[2846],[2830,4070,4071],{"transform":4027},[2835,4072],{"d":4073,"fill":2953,"stroke":2953,"className":4074,"style":4032},"M95.907-62.143L94.684-64.999Q94.602-65.174 94.458-65.219Q94.313-65.264 94.044-65.264L94.044-65.561L95.755-65.561L95.755-65.264Q95.333-65.264 95.333-65.081Q95.333-65.046 95.348-64.999L96.294-62.807L97.134-64.784Q97.173-64.862 97.173-64.952Q97.173-65.092 97.067-65.178Q96.962-65.264 96.821-65.264L96.821-65.561L98.173-65.561L98.173-65.264Q97.649-65.264 97.434-64.784L96.309-62.143Q96.247-62.034 96.141-62.034L96.075-62.034Q95.962-62.034 95.907-62.143",[2846],[2830,4076,4077],{"transform":4027},[2835,4078],{"d":4079,"fill":2953,"stroke":2953,"className":4080,"style":4032},"M98.356-63.866Q98.356-64.346 98.589-64.762Q98.821-65.178 99.231-65.428Q99.641-65.678 100.118-65.678Q100.848-65.678 101.247-65.237Q101.645-64.796 101.645-64.065Q101.645-63.960 101.552-63.936L99.102-63.936L99.102-63.866Q99.102-63.456 99.223-63.100Q99.345-62.745 99.616-62.528Q99.888-62.311 100.317-62.311Q100.680-62.311 100.977-62.540Q101.274-62.768 101.376-63.120Q101.384-63.167 101.470-63.182L101.552-63.182Q101.645-63.155 101.645-63.073Q101.645-63.065 101.638-63.034Q101.575-62.807 101.436-62.624Q101.298-62.440 101.106-62.307Q100.915-62.175 100.696-62.104Q100.477-62.034 100.239-62.034Q99.868-62.034 99.530-62.171Q99.192-62.307 98.925-62.559Q98.657-62.811 98.507-63.151Q98.356-63.491 98.356-63.866M99.110-64.174L101.071-64.174Q101.071-64.479 100.970-64.770Q100.868-65.061 100.651-65.243Q100.434-65.424 100.118-65.424Q99.817-65.424 99.587-65.237Q99.356-65.049 99.233-64.758Q99.110-64.467 99.110-64.174M102.177-62.120L102.177-63.342Q102.177-63.370 102.208-63.401Q102.239-63.432 102.263-63.432L102.368-63.432Q102.438-63.432 102.454-63.370Q102.516-63.050 102.655-62.809Q102.794-62.569 103.026-62.428Q103.259-62.288 103.567-62.288Q103.805-62.288 104.014-62.348Q104.223-62.409 104.360-62.557Q104.497-62.706 104.497-62.952Q104.497-63.206 104.286-63.372Q104.075-63.538 103.805-63.592L103.184-63.706Q102.778-63.784 102.477-64.040Q102.177-64.296 102.177-64.671Q102.177-65.038 102.378-65.260Q102.579-65.483 102.903-65.581Q103.227-65.678 103.567-65.678Q104.032-65.678 104.329-65.471L104.552-65.655Q104.575-65.678 104.606-65.678L104.657-65.678Q104.688-65.678 104.716-65.651Q104.743-65.624 104.743-65.592L104.743-64.608Q104.743-64.577 104.718-64.548Q104.692-64.518 104.657-64.518L104.552-64.518Q104.516-64.518 104.489-64.546Q104.462-64.573 104.462-64.608Q104.462-65.007 104.210-65.227Q103.958-65.448 103.559-65.448Q103.204-65.448 102.921-65.325Q102.638-65.202 102.638-64.897Q102.638-64.678 102.839-64.546Q103.040-64.413 103.286-64.370L103.911-64.257Q104.341-64.167 104.649-63.870Q104.958-63.573 104.958-63.159Q104.958-62.589 104.559-62.311Q104.161-62.034 103.567-62.034Q103.016-62.034 102.665-62.370L102.368-62.057Q102.345-62.034 102.309-62.034L102.263-62.034Q102.239-62.034 102.208-62.065Q102.177-62.096 102.177-62.120",[2846],[3003,4082,4084,4085,4118,4119],{"className":4083},[3006],"Lattice paths on a ",[394,4086,4088],{"className":4087},[397],[394,4089,4091,4109],{"className":4090,"ariaHidden":402},[401],[394,4092,4094,4097,4100,4103,4106],{"className":4093},[406],[394,4095],{"className":4096,"style":3382},[410],[394,4098,3386],{"className":4099},[415],[394,4101],{"className":4102,"style":626},[465],[394,4104,3793],{"className":4105},[516],[394,4107],{"className":4108,"style":626},[465],[394,4110,4112,4115],{"className":4111},[406],[394,4113],{"className":4114,"style":639},[410],[394,4116,3386],{"className":4117},[415]," grid: each node sums its left and top neighbour; corner is ",[394,4120,4122],{"className":4121},[397],[394,4123,4125,4206],{"className":4124,"ariaHidden":402},[401],[394,4126,4128,4131,4197,4200,4203],{"className":4127},[406],[394,4129],{"className":4130,"style":2276},[410],[394,4132,4134,4140,4191],{"className":4133},[415],[394,4135,4137],{"className":4136,"style":1226},[539,1225],[394,4138,540],{"className":4139},[1230,1507],[394,4141,4143],{"className":4142},[909],[394,4144,4146,4183],{"className":4145},[427,913],[394,4147,4149,4180],{"className":4148},[431],[394,4150,4152,4166],{"className":4151,"style":2203},[435],[394,4153,4154,4157],{"style":1523},[394,4155],{"className":4156,"style":443},[442],[394,4158,4160],{"className":4159},[447,448,449,450],[394,4161,4163],{"className":4162},[415,450],[394,4164,520],{"className":4165},[415,450],[394,4167,4168,4171],{"style":1538},[394,4169],{"className":4170,"style":443},[442],[394,4172,4174],{"className":4173},[447,448,449,450],[394,4175,4177],{"className":4176},[415,450],[394,4178,3330],{"className":4179},[415,450],[394,4181,983],{"className":4182},[982],[394,4184,4186],{"className":4185},[431],[394,4187,4189],{"className":4188,"style":1560},[435],[394,4190],{},[394,4192,4194],{"className":4193,"style":1226},[562,1225],[394,4195,563],{"className":4196},[1230,1507],[394,4198],{"className":4199,"style":466},[465],[394,4201,674],{"className":4202},[470],[394,4204],{"className":4205,"style":466},[465],[394,4207,4209,4212],{"className":4208},[406],[394,4210],{"className":4211,"style":639},[410],[394,4213,3366],{"className":4214},[415],[2014,4216,4218],{"id":4217},"the-binomial-theorem","The binomial theorem",[381,4220,4221],{},"The coefficients earn their name from the expansion",[394,4223,4225],{"className":4224},[647],[394,4226,4228],{"className":4227},[397],[394,4229,4231,4253,4304],{"className":4230,"ariaHidden":402},[401],[394,4232,4234,4237,4240,4244,4247,4250],{"className":4233},[406],[394,4235],{"className":4236,"style":535},[410],[394,4238,540],{"className":4239},[539],[394,4241,4243],{"className":4242},[415,419],"x",[394,4245],{"className":4246,"style":626},[465],[394,4248,1080],{"className":4249},[516],[394,4251],{"className":4252,"style":626},[465],[394,4254,4256,4259,4264,4295,4298,4301],{"className":4255},[406],[394,4257],{"className":4258,"style":535},[410],[394,4260,4263],{"className":4261,"style":4262},[415,419],"margin-right:0.0359em;","y",[394,4265,4267,4270],{"className":4266},[562],[394,4268,563],{"className":4269},[562],[394,4271,4273],{"className":4272},[423],[394,4274,4276],{"className":4275},[427],[394,4277,4279],{"className":4278},[431],[394,4280,4283],{"className":4281,"style":4282},[435],"height:0.7144em;",[394,4284,4286,4289],{"style":4285},"top:-3.113em;margin-right:0.05em;",[394,4287],{"className":4288,"style":443},[442],[394,4290,4292],{"className":4291},[447,448,449,450],[394,4293,605],{"className":4294},[415,419,450],[394,4296],{"className":4297,"style":466},[465],[394,4299,674],{"className":4300},[470],[394,4302],{"className":4303,"style":466},[465],[394,4305,4307,4311,4389,4392,4452,4455,4488,4491,4532],{"className":4306},[406],[394,4308],{"className":4309,"style":4310},[410],"height:2.9535em;vertical-align:-1.3021em;",[394,4312,4316],{"className":4313},[4314,4315],"mop","op-limits",[394,4317,4319,4380],{"className":4318},[427,913],[394,4320,4322,4377],{"className":4321},[431],[394,4323,4326,4348,4362],{"className":4324,"style":4325},[435],"height:1.6514em;",[394,4327,4329,4333],{"style":4328},"top:-1.8479em;margin-left:0em;",[394,4330],{"className":4331,"style":4332},[442],"height:3.05em;",[394,4334,4336],{"className":4335},[447,448,449,450],[394,4337,4339,4342,4345],{"className":4338},[415,450],[394,4340,2068],{"className":4341,"style":2067},[415,419,450],[394,4343,674],{"className":4344},[470,450],[394,4346,780],{"className":4347},[415,450],[394,4349,4351,4354],{"style":4350},"top:-3.05em;",[394,4352],{"className":4353,"style":4332},[442],[394,4355,4356],{},[394,4357,4361],{"className":4358},[4314,4359,4360],"op-symbol","large-op","∑",[394,4363,4365,4368],{"style":4364},"top:-4.3em;margin-left:0em;",[394,4366],{"className":4367,"style":4332},[442],[394,4369,4371],{"className":4370},[447,448,449,450],[394,4372,4374],{"className":4373},[415,450],[394,4375,605],{"className":4376},[415,419,450],[394,4378,983],{"className":4379},[982],[394,4381,4383],{"className":4382},[431],[394,4384,4387],{"className":4385,"style":4386},[435],"height:1.3021em;",[394,4388],{},[394,4390],{"className":4391,"style":734},[465],[394,4393,4395,4401,4446],{"className":4394},[415],[394,4396,4398],{"className":4397,"style":1226},[539,1225],[394,4399,540],{"className":4400},[1230,449],[394,4402,4404],{"className":4403},[909],[394,4405,4407,4438],{"className":4406},[427,913],[394,4408,4410,4435],{"className":4409},[431],[394,4411,4413,4424],{"className":4412,"style":1243},[435],[394,4414,4415,4418],{"style":923},[394,4416],{"className":4417,"style":927},[442],[394,4419,4421],{"className":4420},[415],[394,4422,2068],{"className":4423,"style":2067},[415,419],[394,4425,4426,4429],{"style":966},[394,4427],{"className":4428,"style":927},[442],[394,4430,4432],{"className":4431},[415],[394,4433,605],{"className":4434},[415,419],[394,4436,983],{"className":4437},[982],[394,4439,4441],{"className":4440},[431],[394,4442,4444],{"className":4443,"style":1275},[435],[394,4445],{},[394,4447,4449],{"className":4448,"style":1226},[562,1225],[394,4450,563],{"className":4451},[1230,449],[394,4453],{"className":4454,"style":734},[465],[394,4456,4458,4461],{"className":4457},[415],[394,4459,4243],{"className":4460},[415,419],[394,4462,4464],{"className":4463},[423],[394,4465,4467],{"className":4466},[427],[394,4468,4470],{"className":4469},[431],[394,4471,4474],{"className":4472,"style":4473},[435],"height:0.8991em;",[394,4475,4476,4479],{"style":4285},[394,4477],{"className":4478,"style":443},[442],[394,4480,4482],{"className":4481},[447,448,449,450],[394,4483,4485],{"className":4484},[415,450],[394,4486,2068],{"className":4487,"style":2067},[415,419,450],[394,4489],{"className":4490,"style":734},[465],[394,4492,4494,4497],{"className":4493},[415],[394,4495,4263],{"className":4496,"style":4262},[415,419],[394,4498,4500],{"className":4499},[423],[394,4501,4503],{"className":4502},[427],[394,4504,4506],{"className":4505},[431],[394,4507,4509],{"className":4508,"style":4473},[435],[394,4510,4511,4514],{"style":4285},[394,4512],{"className":4513,"style":443},[442],[394,4515,4517],{"className":4516},[447,448,449,450],[394,4518,4520,4523,4526,4529],{"className":4519},[415,450],[394,4521],{"className":4522,"style":3711},[465,450],[394,4524,605],{"className":4525},[415,419,450],[394,4527,457],{"className":4528},[516,450],[394,4530,2068],{"className":4531,"style":2067},[415,419,450],[394,4533,769],{"className":4534},[768],[381,4536,4537,4538,4553,4554,823,4569,4584,4585,4601,4602,4680,4681,4732,4733,4904,4905,4920],{},"since each term of the product picks ",[394,4539,4541],{"className":4540},[397],[394,4542,4544],{"className":4543,"ariaHidden":402},[401],[394,4545,4547,4550],{"className":4546},[406],[394,4548],{"className":4549,"style":601},[410],[394,4551,4243],{"className":4552},[415,419]," from ",[394,4555,4557],{"className":4556},[397],[394,4558,4560],{"className":4559,"ariaHidden":402},[401],[394,4561,4563,4566],{"className":4562},[406],[394,4564],{"className":4565,"style":660},[410],[394,4567,2068],{"className":4568,"style":2067},[415,419],[394,4570,4572],{"className":4571},[397],[394,4573,4575],{"className":4574,"ariaHidden":402},[401],[394,4576,4578,4581],{"className":4577},[406],[394,4579],{"className":4580,"style":601},[410],[394,4582,605],{"className":4583},[415,419]," factors and ",[394,4586,4588],{"className":4587},[397],[394,4589,4591],{"className":4590,"ariaHidden":402},[401],[394,4592,4594,4598],{"className":4593},[406],[394,4595],{"className":4596,"style":4597},[410],"height:0.625em;vertical-align:-0.1944em;",[394,4599,4263],{"className":4600,"style":4262},[415,419]," from the\nrest, and there are ",[394,4603,4605],{"className":4604},[397],[394,4606,4608],{"className":4607,"ariaHidden":402},[401],[394,4609,4611,4614],{"className":4610},[406],[394,4612],{"className":4613,"style":1497},[410],[394,4615,4617,4623,4674],{"className":4616},[415],[394,4618,4620],{"className":4619,"style":1226},[539,1225],[394,4621,540],{"className":4622},[1230,1507],[394,4624,4626],{"className":4625},[909],[394,4627,4629,4666],{"className":4628},[427,913],[394,4630,4632,4663],{"className":4631},[431],[394,4633,4635,4649],{"className":4634,"style":1520},[435],[394,4636,4637,4640],{"style":1523},[394,4638],{"className":4639,"style":443},[442],[394,4641,4643],{"className":4642},[447,448,449,450],[394,4644,4646],{"className":4645},[415,450],[394,4647,2068],{"className":4648,"style":2067},[415,419,450],[394,4650,4651,4654],{"style":1538},[394,4652],{"className":4653,"style":443},[442],[394,4655,4657],{"className":4656},[447,448,449,450],[394,4658,4660],{"className":4659},[415,450],[394,4661,605],{"className":4662},[415,419,450],[394,4664,983],{"className":4665},[982],[394,4667,4669],{"className":4668},[431],[394,4670,4672],{"className":4671,"style":1560},[435],[394,4673],{},[394,4675,4677],{"className":4676,"style":1226},[562,1225],[394,4678,563],{"className":4679},[1230,1507]," ways to make that choice. Setting ",[394,4682,4684],{"className":4683},[397],[394,4685,4687,4705,4723],{"className":4686,"ariaHidden":402},[401],[394,4688,4690,4693,4696,4699,4702],{"className":4689},[406],[394,4691],{"className":4692,"style":601},[410],[394,4694,4243],{"className":4695},[415,419],[394,4697],{"className":4698,"style":466},[465],[394,4700,674],{"className":4701},[470],[394,4703],{"className":4704,"style":466},[465],[394,4706,4708,4711,4714,4717,4720],{"className":4707},[406],[394,4709],{"className":4710,"style":4597},[410],[394,4712,4263],{"className":4713,"style":4262},[415,419],[394,4715],{"className":4716,"style":466},[465],[394,4718,674],{"className":4719},[470],[394,4721],{"className":4722,"style":466},[465],[394,4724,4726,4729],{"className":4725},[406],[394,4727],{"className":4728,"style":639},[410],[394,4730,461],{"className":4731},[415]," gives\n",[394,4734,4736],{"className":4735},[397],[394,4737,4739,4868],{"className":4738,"ariaHidden":402},[401],[394,4740,4742,4745,4790,4793,4859,4862,4865],{"className":4741},[406],[394,4743],{"className":4744,"style":1497},[410],[394,4746,4748,4753],{"className":4747},[4314],[394,4749,4361],{"className":4750,"style":4752},[4314,4359,4751],"small-op","position:relative;top:0em;",[394,4754,4756],{"className":4755},[423],[394,4757,4759,4781],{"className":4758},[427,913],[394,4760,4762,4778],{"className":4761},[431],[394,4763,4766],{"className":4764,"style":4765},[435],"height:0.1864em;",[394,4767,4769,4772],{"style":4768},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[394,4770],{"className":4771,"style":443},[442],[394,4773,4775],{"className":4774},[447,448,449,450],[394,4776,2068],{"className":4777,"style":2067},[415,419,450],[394,4779,983],{"className":4780},[982],[394,4782,4784],{"className":4783},[431],[394,4785,4788],{"className":4786,"style":4787},[435],"height:0.2997em;",[394,4789],{},[394,4791],{"className":4792,"style":734},[465],[394,4794,4796,4802,4853],{"className":4795},[415],[394,4797,4799],{"className":4798,"style":1226},[539,1225],[394,4800,540],{"className":4801},[1230,1507],[394,4803,4805],{"className":4804},[909],[394,4806,4808,4845],{"className":4807},[427,913],[394,4809,4811,4842],{"className":4810},[431],[394,4812,4814,4828],{"className":4813,"style":1520},[435],[394,4815,4816,4819],{"style":1523},[394,4817],{"className":4818,"style":443},[442],[394,4820,4822],{"className":4821},[447,448,449,450],[394,4823,4825],{"className":4824},[415,450],[394,4826,2068],{"className":4827,"style":2067},[415,419,450],[394,4829,4830,4833],{"style":1538},[394,4831],{"className":4832,"style":443},[442],[394,4834,4836],{"className":4835},[447,448,449,450],[394,4837,4839],{"className":4838},[415,450],[394,4840,605],{"className":4841},[415,419,450],[394,4843,983],{"className":4844},[982],[394,4846,4848],{"className":4847},[431],[394,4849,4851],{"className":4850,"style":1560},[435],[394,4852],{},[394,4854,4856],{"className":4855,"style":1226},[562,1225],[394,4857,563],{"className":4858},[1230,1507],[394,4860],{"className":4861,"style":466},[465],[394,4863,674],{"className":4864},[470],[394,4866],{"className":4867,"style":466},[465],[394,4869,4871,4875],{"className":4870},[406],[394,4872],{"className":4873,"style":4874},[410],"height:0.6644em;",[394,4876,4878,4881],{"className":4877},[415],[394,4879,520],{"className":4880},[415],[394,4882,4884],{"className":4883},[423],[394,4885,4887],{"className":4886},[427],[394,4888,4890],{"className":4889},[431],[394,4891,4893],{"className":4892,"style":4874},[435],[394,4894,4895,4898],{"style":438},[394,4896],{"className":4897,"style":443},[442],[394,4899,4901],{"className":4900},[447,448,449,450],[394,4902,605],{"className":4903},[415,419,450],": the number of subsets of an ",[394,4906,4908],{"className":4907},[397],[394,4909,4911],{"className":4910,"ariaHidden":402},[401],[394,4912,4914,4917],{"className":4913},[406],[394,4915],{"className":4916,"style":601},[410],[394,4918,605],{"className":4919},[415,419],"-set, counted by size.",[578,4922,4924],{"id":4923},"combinations-with-repetition-stars-and-bars","Combinations with repetition: stars and bars",[381,4926,4927,4928,4943,4944,4947,4948,4963,4964,5168,5169,5243,5244,5259,5260,5275],{},"How many ways can we write a non-negative integer ",[394,4929,4931],{"className":4930},[397],[394,4932,4934],{"className":4933,"ariaHidden":402},[401],[394,4935,4937,4940],{"className":4936},[406],[394,4938],{"className":4939,"style":601},[410],[394,4941,605],{"className":4942},[415,419]," as an ",[389,4945,4946],{},"ordered"," sum of ",[394,4949,4951],{"className":4950},[397],[394,4952,4954],{"className":4953,"ariaHidden":402},[401],[394,4955,4957,4960],{"className":4956},[406],[394,4958],{"className":4959,"style":660},[410],[394,4961,2068],{"className":4962,"style":2067},[415,419],"\nnon-negative parts, ",[394,4965,4967],{"className":4966},[397],[394,4968,4970,4988,5047,5102,5120],{"className":4969,"ariaHidden":402},[401],[394,4971,4973,4976,4979,4982,4985],{"className":4972},[406],[394,4974],{"className":4975,"style":601},[410],[394,4977,605],{"className":4978},[415,419],[394,4980],{"className":4981,"style":466},[465],[394,4983,674],{"className":4984},[470],[394,4986],{"className":4987,"style":466},[465],[394,4989,4991,4995,5038,5041,5044],{"className":4990},[406],[394,4992],{"className":4993,"style":4994},[410],"height:0.7333em;vertical-align:-0.15em;",[394,4996,4998,5001],{"className":4997},[415],[394,4999,4243],{"className":5000},[415,419],[394,5002,5004],{"className":5003},[423],[394,5005,5007,5029],{"className":5006},[427,913],[394,5008,5010,5026],{"className":5009},[431],[394,5011,5014],{"className":5012,"style":5013},[435],"height:0.3011em;",[394,5015,5017,5020],{"style":5016},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[394,5018],{"className":5019,"style":443},[442],[394,5021,5023],{"className":5022},[447,448,449,450],[394,5024,461],{"className":5025},[415,450],[394,5027,983],{"className":5028},[982],[394,5030,5032],{"className":5031},[431],[394,5033,5036],{"className":5034,"style":5035},[435],"height:0.15em;",[394,5037],{},[394,5039],{"className":5040,"style":626},[465],[394,5042,1080],{"className":5043},[516],[394,5045],{"className":5046,"style":626},[465],[394,5048,5050,5053,5093,5096,5099],{"className":5049},[406],[394,5051],{"className":5052,"style":4994},[410],[394,5054,5056,5059],{"className":5055},[415],[394,5057,4243],{"className":5058},[415,419],[394,5060,5062],{"className":5061},[423],[394,5063,5065,5085],{"className":5064},[427,913],[394,5066,5068,5082],{"className":5067},[431],[394,5069,5071],{"className":5070,"style":5013},[435],[394,5072,5073,5076],{"style":5016},[394,5074],{"className":5075,"style":443},[442],[394,5077,5079],{"className":5078},[447,448,449,450],[394,5080,520],{"className":5081},[415,450],[394,5083,983],{"className":5084},[982],[394,5086,5088],{"className":5087},[431],[394,5089,5091],{"className":5090,"style":5035},[435],[394,5092],{},[394,5094],{"className":5095,"style":626},[465],[394,5097,1080],{"className":5098},[516],[394,5100],{"className":5101,"style":626},[465],[394,5103,5105,5108,5111,5114,5117],{"className":5104},[406],[394,5106],{"className":5107,"style":619},[410],[394,5109,739],{"className":5110},[738],[394,5112],{"className":5113,"style":626},[465],[394,5115,1080],{"className":5116},[516],[394,5118],{"className":5119,"style":626},[465],[394,5121,5123,5127],{"className":5122},[406],[394,5124],{"className":5125,"style":5126},[410],"height:0.5806em;vertical-align:-0.15em;",[394,5128,5130,5133],{"className":5129},[415],[394,5131,4243],{"className":5132},[415,419],[394,5134,5136],{"className":5135},[423],[394,5137,5139,5160],{"className":5138},[427,913],[394,5140,5142,5157],{"className":5141},[431],[394,5143,5146],{"className":5144,"style":5145},[435],"height:0.3361em;",[394,5147,5148,5151],{"style":5016},[394,5149],{"className":5150,"style":443},[442],[394,5152,5154],{"className":5153},[447,448,449,450],[394,5155,2068],{"className":5156,"style":2067},[415,419,450],[394,5158,983],{"className":5159},[982],[394,5161,5163],{"className":5162},[431],[394,5164,5166],{"className":5165,"style":5035},[435],[394,5167],{}," with each ",[394,5170,5172],{"className":5171},[397],[394,5173,5175,5234],{"className":5174,"ariaHidden":402},[401],[394,5176,5178,5182,5224,5227,5231],{"className":5177},[406],[394,5179],{"className":5180,"style":5181},[410],"height:0.786em;vertical-align:-0.15em;",[394,5183,5185,5188],{"className":5184},[415],[394,5186,4243],{"className":5187},[415,419],[394,5189,5191],{"className":5190},[423],[394,5192,5194,5216],{"className":5193},[427,913],[394,5195,5197,5213],{"className":5196},[431],[394,5198,5201],{"className":5199,"style":5200},[435],"height:0.3117em;",[394,5202,5203,5206],{"style":5016},[394,5204],{"className":5205,"style":443},[442],[394,5207,5209],{"className":5208},[447,448,449,450],[394,5210,5212],{"className":5211},[415,419,450],"i",[394,5214,983],{"className":5215},[982],[394,5217,5219],{"className":5218},[431],[394,5220,5222],{"className":5221,"style":5035},[435],[394,5223],{},[394,5225],{"className":5226,"style":466},[465],[394,5228,5230],{"className":5229},[470],"≥",[394,5232],{"className":5233,"style":466},[465],[394,5235,5237,5240],{"className":5236},[406],[394,5238],{"className":5239,"style":639},[410],[394,5241,780],{"className":5242},[415],"? Equivalently,\nhow many multisets of size ",[394,5245,5247],{"className":5246},[397],[394,5248,5250],{"className":5249,"ariaHidden":402},[401],[394,5251,5253,5256],{"className":5252},[406],[394,5254],{"className":5255,"style":601},[410],[394,5257,605],{"className":5258},[415,419]," can we draw from ",[394,5261,5263],{"className":5262},[397],[394,5264,5266],{"className":5265,"ariaHidden":402},[401],[394,5267,5269,5272],{"className":5268},[406],[394,5270],{"className":5271,"style":660},[410],[394,5273,2068],{"className":5274,"style":2067},[415,419]," distinct types?",[2022,5277,5278],{"type":2024},[381,5279,5280,5283,5284],{},[389,5281,5282],{},"Lemma (Stars and bars)."," The number of such solutions is\n",[394,5285,5287],{"className":5286},[397],[394,5288,5290],{"className":5289,"ariaHidden":402},[401],[394,5291,5293,5297,5388],{"className":5292},[406],[394,5294],{"className":5295,"style":5296},[410],"height:1.3334em;vertical-align:-0.4033em;",[394,5298,5300,5306,5382],{"className":5299},[415],[394,5301,5303],{"className":5302,"style":1226},[539,1225],[394,5304,540],{"className":5305},[1230,1507],[394,5307,5309],{"className":5308},[909],[394,5310,5312,5374],{"className":5311},[427,913],[394,5313,5315,5371],{"className":5314},[431],[394,5316,5319,5345],{"className":5317,"style":5318},[435],"height:0.9301em;",[394,5320,5321,5324],{"style":1523},[394,5322],{"className":5323,"style":443},[442],[394,5325,5327],{"className":5326},[447,448,449,450],[394,5328,5330,5333,5336,5339,5342],{"className":5329},[415,450],[394,5331],{"className":5332,"style":3711},[465,450],[394,5334,2068],{"className":5335,"style":2067},[415,419,450],[394,5337,457],{"className":5338},[516,450],[394,5340,461],{"className":5341},[415,450],[394,5343],{"className":5344,"style":3711},[465,450],[394,5346,5347,5350],{"style":1538},[394,5348],{"className":5349,"style":443},[442],[394,5351,5353],{"className":5352},[447,448,449,450],[394,5354,5356,5359,5362,5365,5368],{"className":5355},[415,450],[394,5357,605],{"className":5358},[415,419,450],[394,5360,1080],{"className":5361},[516,450],[394,5363,2068],{"className":5364,"style":2067},[415,419,450],[394,5366,457],{"className":5367},[516,450],[394,5369,461],{"className":5370},[415,450],[394,5372,983],{"className":5373},[982],[394,5375,5377],{"className":5376},[431],[394,5378,5380],{"className":5379,"style":1776},[435],[394,5381],{},[394,5383,5385],{"className":5384,"style":1226},[562,1225],[394,5386,563],{"className":5387},[1230,1507],[394,5389,1099],{"className":5390},[415],[381,5392,5393,5396,5397,5412,5413,5416,5417,392,5450,5453,5454,5469,5470,5486,5487,5539,5540,5574,5575,5608],{},[1107,5394,5395],{},"Bijection."," Lay out ",[394,5398,5400],{"className":5399},[397],[394,5401,5403],{"className":5402,"ariaHidden":402},[401],[394,5404,5406,5409],{"className":5405},[406],[394,5407],{"className":5408,"style":601},[410],[394,5410,605],{"className":5411},[415,419]," identical ",[389,5414,5415],{},"stars"," in a row and insert ",[394,5418,5420],{"className":5419},[397],[394,5421,5423,5441],{"className":5422,"ariaHidden":402},[401],[394,5424,5426,5429,5432,5435,5438],{"className":5425},[406],[394,5427],{"className":5428,"style":2487},[410],[394,5430,2068],{"className":5431,"style":2067},[415,419],[394,5433],{"className":5434,"style":626},[465],[394,5436,457],{"className":5437},[516],[394,5439],{"className":5440,"style":626},[465],[394,5442,5444,5447],{"className":5443},[406],[394,5445],{"className":5446,"style":639},[410],[394,5448,461],{"className":5449},[415],[389,5451,5452],{},"bars"," among\nthem; the bars cut the stars into ",[394,5455,5457],{"className":5456},[397],[394,5458,5460],{"className":5459,"ariaHidden":402},[401],[394,5461,5463,5466],{"className":5462},[406],[394,5464],{"className":5465,"style":660},[410],[394,5467,2068],{"className":5468,"style":2067},[415,419]," ordered groups, the ",[394,5471,5473],{"className":5472},[397],[394,5474,5476],{"className":5475,"ariaHidden":402},[401],[394,5477,5479,5483],{"className":5478},[406],[394,5480],{"className":5481,"style":5482},[410],"height:0.6595em;",[394,5484,5212],{"className":5485},[415,419],"-th group's size being\n",[394,5488,5490],{"className":5489},[397],[394,5491,5493],{"className":5492,"ariaHidden":402},[401],[394,5494,5496,5499],{"className":5495},[406],[394,5497],{"className":5498,"style":5126},[410],[394,5500,5502,5505],{"className":5501},[415],[394,5503,4243],{"className":5504},[415,419],[394,5506,5508],{"className":5507},[423],[394,5509,5511,5531],{"className":5510},[427,913],[394,5512,5514,5528],{"className":5513},[431],[394,5515,5517],{"className":5516,"style":5200},[435],[394,5518,5519,5522],{"style":5016},[394,5520],{"className":5521,"style":443},[442],[394,5523,5525],{"className":5524},[447,448,449,450],[394,5526,5212],{"className":5527},[415,419,450],[394,5529,983],{"className":5530},[982],[394,5532,5534],{"className":5533},[431],[394,5535,5537],{"className":5536,"style":5035},[435],[394,5538],{},". For example with ",[394,5541,5543],{"className":5542},[397],[394,5544,5546,5564],{"className":5545,"ariaHidden":402},[401],[394,5547,5549,5552,5555,5558,5561],{"className":5548},[406],[394,5550],{"className":5551,"style":601},[410],[394,5553,605],{"className":5554},[415,419],[394,5556],{"className":5557,"style":466},[465],[394,5559,674],{"className":5560},[470],[394,5562],{"className":5563,"style":466},[465],[394,5565,5567,5570],{"className":5566},[406],[394,5568],{"className":5569,"style":639},[410],[394,5571,5573],{"className":5572},[415],"5",", ",[394,5576,5578],{"className":5577},[397],[394,5579,5581,5599],{"className":5580,"ariaHidden":402},[401],[394,5582,5584,5587,5590,5593,5596],{"className":5583},[406],[394,5585],{"className":5586,"style":660},[410],[394,5588,2068],{"className":5589,"style":2067},[415,419],[394,5591],{"className":5592,"style":466},[465],[394,5594,674],{"className":5595},[470],[394,5597],{"className":5598,"style":466},[465],[394,5600,5602,5605],{"className":5601},[406],[394,5603],{"className":5604,"style":639},[410],[394,5606,3386],{"className":5607},[415],":",[394,5610,5612],{"className":5611},[647],[394,5613,5615],{"className":5614},[397],[394,5616,5618,5644,5656,5681,5708,5763,5831,5898],{"className":5617,"ariaHidden":402},[401],[394,5619,5621,5624,5628,5631,5634,5637,5641],{"className":5620},[406],[394,5622],{"className":5623,"style":535},[410],[394,5625,5627],{"className":5626},[415],"⋆",[394,5629],{"className":5630,"style":734},[465],[394,5632,5627],{"className":5633},[415],[394,5635],{"className":5636,"style":466},[465],[394,5638,5640],{"className":5639},[470],"∣",[394,5642],{"className":5643,"style":466},[465],[394,5645,5647,5650,5653],{"className":5646},[406],[394,5648],{"className":5649,"style":535},[410],[394,5651,5640],{"className":5652},[470],[394,5654],{"className":5655,"style":466},[465],[394,5657,5659,5663,5666,5669,5672,5675,5678],{"className":5658},[406],[394,5660],{"className":5661,"style":5662},[410],"height:0.4653em;",[394,5664,5627],{"className":5665},[415],[394,5667],{"className":5668,"style":734},[465],[394,5670],{"className":5671,"style":626},[465],[394,5673,5627],{"className":5674},[516],[394,5676],{"className":5677,"style":734},[465],[394,5679],{"className":5680,"style":626},[465],[394,5682,5684,5688,5691,5695,5698,5702,5705],{"className":5683},[406],[394,5685],{"className":5686,"style":5687},[410],"height:0.522em;vertical-align:-0.011em;",[394,5689,5627],{"className":5690},[415],[394,5692],{"className":5693,"style":5694},[465],"margin-right:1em;",[394,5696],{"className":5697,"style":466},[465],[394,5699,5701],{"className":5700},[470],"⟷",[394,5703],{"className":5704,"style":5694},[465],[394,5706],{"className":5707,"style":466},[465],[394,5709,5711,5714,5754,5757,5760],{"className":5710},[406],[394,5712],{"className":5713,"style":5126},[410],[394,5715,5717,5720],{"className":5716},[415],[394,5718,4243],{"className":5719},[415,419],[394,5721,5723],{"className":5722},[423],[394,5724,5726,5746],{"className":5725},[427,913],[394,5727,5729,5743],{"className":5728},[431],[394,5730,5732],{"className":5731,"style":5013},[435],[394,5733,5734,5737],{"style":5016},[394,5735],{"className":5736,"style":443},[442],[394,5738,5740],{"className":5739},[447,448,449,450],[394,5741,461],{"className":5742},[415,450],[394,5744,983],{"className":5745},[982],[394,5747,5749],{"className":5748},[431],[394,5750,5752],{"className":5751,"style":5035},[435],[394,5753],{},[394,5755],{"className":5756,"style":466},[465],[394,5758,674],{"className":5759},[470],[394,5761],{"className":5762,"style":466},[465],[394,5764,5766,5770,5773,5776,5779,5782,5822,5825,5828],{"className":5765},[406],[394,5767],{"className":5768,"style":5769},[410],"height:0.8389em;vertical-align:-0.1944em;",[394,5771,520],{"className":5772},[415],[394,5774,769],{"className":5775},[768],[394,5777],{"className":5778,"style":466},[465],[394,5780],{"className":5781,"style":734},[465],[394,5783,5785,5788],{"className":5784},[415],[394,5786,4243],{"className":5787},[415,419],[394,5789,5791],{"className":5790},[423],[394,5792,5794,5814],{"className":5793},[427,913],[394,5795,5797,5811],{"className":5796},[431],[394,5798,5800],{"className":5799,"style":5013},[435],[394,5801,5802,5805],{"style":5016},[394,5803],{"className":5804,"style":443},[442],[394,5806,5808],{"className":5807},[447,448,449,450],[394,5809,520],{"className":5810},[415,450],[394,5812,983],{"className":5813},[982],[394,5815,5817],{"className":5816},[431],[394,5818,5820],{"className":5819,"style":5035},[435],[394,5821],{},[394,5823],{"className":5824,"style":466},[465],[394,5826,674],{"className":5827},[470],[394,5829],{"className":5830,"style":466},[465],[394,5832,5834,5837,5840,5843,5846,5849,5889,5892,5895],{"className":5833},[406],[394,5835],{"className":5836,"style":5769},[410],[394,5838,780],{"className":5839},[415],[394,5841,769],{"className":5842},[768],[394,5844],{"className":5845,"style":466},[465],[394,5847],{"className":5848,"style":734},[465],[394,5850,5852,5855],{"className":5851},[415],[394,5853,4243],{"className":5854},[415,419],[394,5856,5858],{"className":5857},[423],[394,5859,5861,5881],{"className":5860},[427,913],[394,5862,5864,5878],{"className":5863},[431],[394,5865,5867],{"className":5866,"style":5013},[435],[394,5868,5869,5872],{"style":5016},[394,5870],{"className":5871,"style":443},[442],[394,5873,5875],{"className":5874},[447,448,449,450],[394,5876,3386],{"className":5877},[415,450],[394,5879,983],{"className":5880},[982],[394,5882,5884],{"className":5883},[431],[394,5885,5887],{"className":5886,"style":5035},[435],[394,5888],{},[394,5890],{"className":5891,"style":466},[465],[394,5893,674],{"className":5894},[470],[394,5896],{"className":5897,"style":466},[465],[394,5899,5901,5904],{"className":5900},[406],[394,5902],{"className":5903,"style":639},[410],[394,5905,5907],{"className":5906},[415],"3.",[381,5909,5910,5911,5926,5927,5960,5961,6012,6013,823,6046,6097,6098,6194,6195],{},"Every arrangement of ",[394,5912,5914],{"className":5913},[397],[394,5915,5917],{"className":5916,"ariaHidden":402},[401],[394,5918,5920,5923],{"className":5919},[406],[394,5921],{"className":5922,"style":601},[410],[394,5924,605],{"className":5925},[415,419]," stars and ",[394,5928,5930],{"className":5929},[397],[394,5931,5933,5951],{"className":5932,"ariaHidden":402},[401],[394,5934,5936,5939,5942,5945,5948],{"className":5935},[406],[394,5937],{"className":5938,"style":2487},[410],[394,5940,2068],{"className":5941,"style":2067},[415,419],[394,5943],{"className":5944,"style":626},[465],[394,5946,457],{"className":5947},[516],[394,5949],{"className":5950,"style":626},[465],[394,5952,5954,5957],{"className":5953},[406],[394,5955],{"className":5956,"style":639},[410],[394,5958,461],{"className":5959},[415]," bars in a line of ",[394,5962,5964],{"className":5963},[397],[394,5965,5967,5985,6003],{"className":5966,"ariaHidden":402},[401],[394,5968,5970,5973,5976,5979,5982],{"className":5969},[406],[394,5971],{"className":5972,"style":619},[410],[394,5974,605],{"className":5975},[415,419],[394,5977],{"className":5978,"style":626},[465],[394,5980,1080],{"className":5981},[516],[394,5983],{"className":5984,"style":626},[465],[394,5986,5988,5991,5994,5997,6000],{"className":5987},[406],[394,5989],{"className":5990,"style":2487},[410],[394,5992,2068],{"className":5993,"style":2067},[415,419],[394,5995],{"className":5996,"style":626},[465],[394,5998,457],{"className":5999},[516],[394,6001],{"className":6002,"style":626},[465],[394,6004,6006,6009],{"className":6005},[406],[394,6007],{"className":6008,"style":639},[410],[394,6010,461],{"className":6011},[415]," symbols yields\nexactly one solution and vice versa, so the count is the number of ways to choose\nwhich ",[394,6014,6016],{"className":6015},[397],[394,6017,6019,6037],{"className":6018,"ariaHidden":402},[401],[394,6020,6022,6025,6028,6031,6034],{"className":6021},[406],[394,6023],{"className":6024,"style":2487},[410],[394,6026,2068],{"className":6027,"style":2067},[415,419],[394,6029],{"className":6030,"style":626},[465],[394,6032,457],{"className":6033},[516],[394,6035],{"className":6036,"style":626},[465],[394,6038,6040,6043],{"className":6039},[406],[394,6041],{"className":6042,"style":639},[410],[394,6044,461],{"className":6045},[415],[394,6047,6049],{"className":6048},[397],[394,6050,6052,6070,6088],{"className":6051,"ariaHidden":402},[401],[394,6053,6055,6058,6061,6064,6067],{"className":6054},[406],[394,6056],{"className":6057,"style":619},[410],[394,6059,605],{"className":6060},[415,419],[394,6062],{"className":6063,"style":626},[465],[394,6065,1080],{"className":6066},[516],[394,6068],{"className":6069,"style":626},[465],[394,6071,6073,6076,6079,6082,6085],{"className":6072},[406],[394,6074],{"className":6075,"style":2487},[410],[394,6077,2068],{"className":6078,"style":2067},[415,419],[394,6080],{"className":6081,"style":626},[465],[394,6083,457],{"className":6084},[516],[394,6086],{"className":6087,"style":626},[465],[394,6089,6091,6094],{"className":6090},[406],[394,6092],{"className":6093,"style":639},[410],[394,6095,461],{"className":6096},[415]," positions hold bars, namely ",[394,6099,6101],{"className":6100},[397],[394,6102,6104],{"className":6103,"ariaHidden":402},[401],[394,6105,6107,6110],{"className":6106},[406],[394,6108],{"className":6109,"style":5296},[410],[394,6111,6113,6119,6188],{"className":6112},[415],[394,6114,6116],{"className":6115,"style":1226},[539,1225],[394,6117,540],{"className":6118},[1230,1507],[394,6120,6122],{"className":6121},[909],[394,6123,6125,6180],{"className":6124},[427,913],[394,6126,6128,6177],{"className":6127},[431],[394,6129,6131,6151],{"className":6130,"style":5318},[435],[394,6132,6133,6136],{"style":1523},[394,6134],{"className":6135,"style":443},[442],[394,6137,6139],{"className":6138},[447,448,449,450],[394,6140,6142,6145,6148],{"className":6141},[415,450],[394,6143,2068],{"className":6144,"style":2067},[415,419,450],[394,6146,457],{"className":6147},[516,450],[394,6149,461],{"className":6150},[415,450],[394,6152,6153,6156],{"style":1538},[394,6154],{"className":6155,"style":443},[442],[394,6157,6159],{"className":6158},[447,448,449,450],[394,6160,6162,6165,6168,6171,6174],{"className":6161},[415,450],[394,6163,605],{"className":6164},[415,419,450],[394,6166,1080],{"className":6167},[516,450],[394,6169,2068],{"className":6170,"style":2067},[415,419,450],[394,6172,457],{"className":6173},[516,450],[394,6175,461],{"className":6176},[415,450],[394,6178,983],{"className":6179},[982],[394,6181,6183],{"className":6182},[431],[394,6184,6186],{"className":6185,"style":1776},[435],[394,6187],{},[394,6189,6191],{"className":6190,"style":1226},[562,1225],[394,6192,563],{"className":6193},[1230,1507],".\n",[394,6196,6198],{"className":6197},[397],[394,6199,6201],{"className":6200,"ariaHidden":402},[401],[394,6202,6204,6207],{"className":6203},[406],[394,6205],{"className":6206,"style":2798},[410],[394,6208,6210],{"className":6209},[2802,2803],[394,6211,2808],{"className":6212},[415,2807],[381,6214,6215,6216,6249,6250,6283,6284,6354,6355,6370,6371,1099],{},"Concretely, with ",[394,6217,6219],{"className":6218},[397],[394,6220,6222,6240],{"className":6221,"ariaHidden":402},[401],[394,6223,6225,6228,6231,6234,6237],{"className":6224},[406],[394,6226],{"className":6227,"style":601},[410],[394,6229,605],{"className":6230},[415,419],[394,6232],{"className":6233,"style":466},[465],[394,6235,674],{"className":6236},[470],[394,6238],{"className":6239,"style":466},[465],[394,6241,6243,6246],{"className":6242},[406],[394,6244],{"className":6245,"style":639},[410],[394,6247,5573],{"className":6248},[415]," and ",[394,6251,6253],{"className":6252},[397],[394,6254,6256,6274],{"className":6255,"ariaHidden":402},[401],[394,6257,6259,6262,6265,6268,6271],{"className":6258},[406],[394,6260],{"className":6261,"style":660},[410],[394,6263,2068],{"className":6264,"style":2067},[415,419],[394,6266],{"className":6267,"style":466},[465],[394,6269,674],{"className":6270},[470],[394,6272],{"className":6273,"style":466},[465],[394,6275,6277,6280],{"className":6276},[406],[394,6278],{"className":6279,"style":639},[410],[394,6281,3386],{"className":6282},[415]," there are ",[394,6285,6287],{"className":6286},[397],[394,6288,6290,6308,6326,6344],{"className":6289,"ariaHidden":402},[401],[394,6291,6293,6296,6299,6302,6305],{"className":6292},[406],[394,6294],{"className":6295,"style":619},[410],[394,6297,605],{"className":6298},[415,419],[394,6300],{"className":6301,"style":626},[465],[394,6303,1080],{"className":6304},[516],[394,6306],{"className":6307,"style":626},[465],[394,6309,6311,6314,6317,6320,6323],{"className":6310},[406],[394,6312],{"className":6313,"style":2487},[410],[394,6315,2068],{"className":6316,"style":2067},[415,419],[394,6318],{"className":6319,"style":626},[465],[394,6321,457],{"className":6322},[516],[394,6324],{"className":6325,"style":626},[465],[394,6327,6329,6332,6335,6338,6341],{"className":6328},[406],[394,6330],{"className":6331,"style":639},[410],[394,6333,461],{"className":6334},[415],[394,6336],{"className":6337,"style":466},[465],[394,6339,674],{"className":6340},[470],[394,6342],{"className":6343,"style":466},[465],[394,6345,6347,6350],{"className":6346},[406],[394,6348],{"className":6349,"style":639},[410],[394,6351,6353],{"className":6352},[415],"7"," slots; choosing the ",[394,6356,6358],{"className":6357},[397],[394,6359,6361],{"className":6360,"ariaHidden":402},[401],[394,6362,6364,6367],{"className":6363},[406],[394,6365],{"className":6366,"style":639},[410],[394,6368,520],{"className":6369},[415]," of\nthem that hold bars fixes the three part sizes at once, so the count is\n",[394,6372,6374],{"className":6373},[397],[394,6375,6377,6458],{"className":6376,"ariaHidden":402},[401],[394,6378,6380,6383,6449,6452,6455],{"className":6379},[406],[394,6381],{"className":6382,"style":2276},[410],[394,6384,6386,6392,6443],{"className":6385},[415],[394,6387,6389],{"className":6388,"style":1226},[539,1225],[394,6390,540],{"className":6391},[1230,1507],[394,6393,6395],{"className":6394},[909],[394,6396,6398,6435],{"className":6397},[427,913],[394,6399,6401,6432],{"className":6400},[431],[394,6402,6404,6418],{"className":6403,"style":2203},[435],[394,6405,6406,6409],{"style":1523},[394,6407],{"className":6408,"style":443},[442],[394,6410,6412],{"className":6411},[447,448,449,450],[394,6413,6415],{"className":6414},[415,450],[394,6416,520],{"className":6417},[415,450],[394,6419,6420,6423],{"style":1538},[394,6421],{"className":6422,"style":443},[442],[394,6424,6426],{"className":6425},[447,448,449,450],[394,6427,6429],{"className":6428},[415,450],[394,6430,6353],{"className":6431},[415,450],[394,6433,983],{"className":6434},[982],[394,6436,6438],{"className":6437},[431],[394,6439,6441],{"className":6440,"style":1560},[435],[394,6442],{},[394,6444,6446],{"className":6445,"style":1226},[562,1225],[394,6447,563],{"className":6448},[1230,1507],[394,6450],{"className":6451,"style":466},[465],[394,6453,674],{"className":6454},[470],[394,6456],{"className":6457,"style":466},[465],[394,6459,6461,6464],{"className":6460},[406],[394,6462],{"className":6463,"style":639},[410],[394,6465,6467],{"className":6466},[415],"21",[2817,6469,6471,6714],{"className":6470},[2820,2821],[2823,6472,6476],{"xmlns":2825,"width":6473,"height":6474,"viewBox":6475},"253.544","108.412","-75 -75 190.158 81.309",[2830,6477,6478,6481,6488,6494,6500,6506,6512,6515,6517,6524,6527,6529,6535,6538,6566,6591,6594,6619],{"stroke":2832,"style":2833},[2835,6479],{"fill":2837,"d":6480},"M-65.203-20.073v-23.046h23.046v23.046ZM-39.596-20.073v-23.046h23.047v23.046ZM-13.988-20.073v-23.046H9.058v23.046ZM11.62-20.073v-23.046h23.046v23.046ZM37.227-20.073v-23.046h23.046v23.046ZM62.834-20.073v-23.046h23.047v23.046ZM88.441-20.073v-23.046h23.047v23.046Zm23.047-23.046",[2830,6482,6484],{"transform":6483},"translate(23.295 2.094)",[2835,6485],{"d":6486,"fill":2832,"stroke":2832,"className":6487,"style":2847},"M-78.405-31.829Q-78.400-31.842-78.398-31.853Q-78.396-31.864-78.396-31.873L-77.526-33.459L-79.143-34.224Q-79.218-34.255-79.218-34.325Q-79.218-34.373-79.176-34.406Q-79.134-34.439-79.090-34.439L-77.315-34.096L-77.091-35.881Q-77.073-35.977-76.976-35.977Q-76.884-35.977-76.867-35.881L-76.642-34.096L-74.867-34.439Q-74.819-34.439-74.781-34.409Q-74.744-34.378-74.744-34.325Q-74.744-34.255-74.806-34.224L-76.436-33.459L-75.570-31.886Q-75.553-31.842-75.553-31.829Q-75.553-31.723-75.671-31.723Q-75.715-31.723-75.759-31.767L-76.976-33.073L-78.198-31.767Q-78.242-31.723-78.290-31.723Q-78.405-31.723-78.405-31.829",[2846],[2830,6489,6491],{"transform":6490},"translate(48.902 2.094)",[2835,6492],{"d":6486,"fill":2832,"stroke":2832,"className":6493,"style":2847},[2846],[2830,6495,6497],{"transform":6496},"translate(125.725 2.094)",[2835,6498],{"d":6486,"fill":2832,"stroke":2832,"className":6499,"style":2847},[2846],[2830,6501,6503],{"transform":6502},"translate(151.332 2.094)",[2835,6504],{"d":6486,"fill":2832,"stroke":2832,"className":6505,"style":2847},[2846],[2830,6507,6509],{"transform":6508},"translate(176.94 2.094)",[2835,6510],{"d":6486,"fill":2832,"stroke":2832,"className":6511,"style":2847},[2846],[2835,6513],{"fill":2952,"stroke":2837,"d":6514},"M-13.988-20.073v-23.046H9.058v23.046ZM9.058-43.119",[2835,6516],{"fill":2837,"stroke":2953,"d":6514,"style":2954},[2830,6518,6520],{"transform":6519},"translate(75.538 2.25)",[2835,6521],{"d":6522,"fill":2832,"stroke":2832,"className":6523,"style":2847},"M-78.198-29.517L-78.198-38.175Q-78.167-38.346-78-38.346Q-77.926-38.346-77.875-38.298Q-77.825-38.249-77.811-38.175L-77.811-29.517Q-77.825-29.443-77.877-29.394Q-77.930-29.346-78-29.346Q-78.167-29.346-78.198-29.517",[2846],[2835,6525],{"fill":2952,"stroke":2837,"d":6526},"M11.62-20.073v-23.046h23.046v23.046Zm23.046-23.046",[2835,6528],{"fill":2837,"stroke":2953,"d":6526,"style":2954},[2830,6530,6532],{"transform":6531},"translate(101.145 2.25)",[2835,6533],{"d":6522,"fill":2832,"stroke":2832,"className":6534,"style":2847},[2846],[2835,6536],{"fill":2837,"d":6537,"style":2986},"M-65.203-14.951h48.654",[2830,6539,6540,6547,6554,6560],{"stroke":2837},[2830,6541,6543],{"transform":6542},"translate(28.505 30.19)",[2835,6544],{"d":6545,"fill":2832,"stroke":2832,"className":6546,"style":4032},"M-78.600-31.885Q-78.433-31.772-78.190-31.772Q-77.940-31.772-77.743-31.998Q-77.546-32.225-77.487-32.483L-77.128-33.924Q-77.050-34.229-77.050-34.389Q-77.050-34.600-77.167-34.735Q-77.284-34.869-77.495-34.869Q-77.749-34.869-77.977-34.723Q-78.206-34.576-78.362-34.346Q-78.518-34.116-78.577-33.869Q-78.589-33.795-78.655-33.795L-78.761-33.795Q-78.792-33.795-78.819-33.830Q-78.847-33.866-78.847-33.893L-78.847-33.924Q-78.768-34.237-78.569-34.510Q-78.370-34.784-78.081-34.953Q-77.792-35.123-77.479-35.123Q-77.183-35.123-76.923-34.979Q-76.663-34.834-76.554-34.573Q-76.405-34.815-76.186-34.969Q-75.968-35.123-75.714-35.123Q-75.515-35.123-75.327-35.057Q-75.140-34.991-75.018-34.852Q-74.897-34.713-74.897-34.518Q-74.897-34.307-75.028-34.151Q-75.159-33.994-75.366-33.994Q-75.499-33.994-75.593-34.078Q-75.686-34.162-75.686-34.299Q-75.686-34.463-75.579-34.592Q-75.472-34.721-75.311-34.756Q-75.487-34.869-75.729-34.869Q-75.897-34.869-76.044-34.760Q-76.190-34.651-76.290-34.487Q-76.390-34.323-76.433-34.155L-76.792-32.717Q-76.862-32.373-76.862-32.252Q-76.862-32.037-76.745-31.905Q-76.628-31.772-76.417-31.772Q-76.038-31.772-75.737-32.078Q-75.436-32.385-75.343-32.772Q-75.315-32.842-75.257-32.842L-75.151-32.842Q-75.112-32.842-75.089-32.813Q-75.065-32.784-75.065-32.748Q-75.065-32.733-75.073-32.717Q-75.151-32.405-75.350-32.131Q-75.550-31.858-75.835-31.688Q-76.120-31.518-76.433-31.518Q-76.733-31.518-76.993-31.662Q-77.253-31.807-77.366-32.069Q-77.511-31.834-77.727-31.676Q-77.944-31.518-78.198-31.518Q-78.397-31.518-78.585-31.584Q-78.772-31.651-78.893-31.789Q-79.015-31.928-79.015-32.123Q-79.015-32.334-78.882-32.489Q-78.749-32.643-78.546-32.643Q-78.401-32.643-78.313-32.561Q-78.225-32.479-78.225-32.338Q-78.225-32.178-78.331-32.049Q-78.436-31.920-78.600-31.885",[2846],[2830,6548,6549],{"transform":6542},[2835,6550],{"d":6551,"fill":2832,"stroke":2832,"className":6552,"style":6553},"M-71.462-30.485L-73.753-30.485L-73.753-30.743Q-72.877-30.743-72.877-30.916L-72.877-33.995Q-73.070-33.907-73.302-33.870Q-73.533-33.834-73.788-33.834L-73.788-34.091Q-73.410-34.091-73.089-34.176Q-72.769-34.261-72.540-34.475L-72.420-34.475Q-72.388-34.475-72.363-34.452Q-72.338-34.428-72.338-34.390L-72.338-30.916Q-72.338-30.743-71.462-30.743",[2846],"stroke-width:0.180",[2830,6555,6556],{"transform":6542},[2835,6557],{"d":6558,"fill":2832,"stroke":2832,"className":6559,"style":4032},"M-64.375-32.573L-69.688-32.573Q-69.766-32.580-69.815-32.629Q-69.863-32.678-69.863-32.756Q-69.863-32.826-69.816-32.877Q-69.770-32.928-69.688-32.940L-64.375-32.940Q-64.301-32.928-64.254-32.877Q-64.207-32.826-64.207-32.756Q-64.207-32.678-64.256-32.629Q-64.305-32.580-64.375-32.573M-64.375-34.260L-69.688-34.260Q-69.766-34.268-69.815-34.317Q-69.863-34.366-69.863-34.444Q-69.863-34.514-69.816-34.565Q-69.770-34.616-69.688-34.627L-64.375-34.627Q-64.301-34.616-64.254-34.565Q-64.207-34.514-64.207-34.444Q-64.207-34.366-64.256-34.317Q-64.305-34.268-64.375-34.260",[2846],[2830,6561,6562],{"transform":6542},[2835,6563],{"d":6564,"fill":2832,"stroke":2832,"className":6565,"style":4032},"M-60.139-31.596L-63.299-31.596L-63.299-31.803Q-63.299-31.830-63.276-31.862L-61.924-33.260Q-61.545-33.647-61.297-33.936Q-61.049-34.225-60.875-34.582Q-60.702-34.940-60.702-35.330Q-60.702-35.678-60.834-35.971Q-60.967-36.264-61.221-36.442Q-61.475-36.619-61.830-36.619Q-62.190-36.619-62.481-36.424Q-62.772-36.229-62.916-35.901L-62.862-35.901Q-62.678-35.901-62.553-35.780Q-62.428-35.659-62.428-35.467Q-62.428-35.287-62.553-35.159Q-62.678-35.030-62.862-35.030Q-63.041-35.030-63.170-35.159Q-63.299-35.287-63.299-35.467Q-63.299-35.869-63.079-36.205Q-62.858-36.541-62.493-36.729Q-62.127-36.916-61.725-36.916Q-61.245-36.916-60.829-36.729Q-60.413-36.541-60.161-36.180Q-59.909-35.819-59.909-35.330Q-59.909-34.971-60.063-34.668Q-60.217-34.366-60.469-34.106Q-60.721-33.846-61.071-33.561Q-61.420-33.276-61.588-33.123L-62.518-32.284L-61.803-32.284Q-60.428-32.284-60.389-32.323Q-60.319-32.401-60.276-32.586Q-60.233-32.772-60.190-33.061L-59.909-33.061",[2846],[2830,6567,6568,6574,6580,6585],{"stroke":2837},[2830,6569,6571],{"transform":6570},"translate(79.72 30.19)",[2835,6572],{"d":6545,"fill":2832,"stroke":2832,"className":6573,"style":4032},[2846],[2830,6575,6576],{"transform":6570},[2835,6577],{"d":6578,"fill":2832,"stroke":2832,"className":6579,"style":6553},"M-71.462-30.485L-74.072-30.485L-74.072-30.670Q-74.066-30.693-74.046-30.719L-72.895-31.774Q-72.555-32.085-72.375-32.271Q-72.194-32.457-72.049-32.717Q-71.904-32.978-71.904-33.274Q-71.904-33.547-72.030-33.762Q-72.156-33.977-72.376-34.097Q-72.596-34.217-72.871-34.217Q-73.047-34.217-73.217-34.160Q-73.387-34.103-73.519-33.996Q-73.650-33.889-73.730-33.731Q-73.642-33.731-73.564-33.687Q-73.486-33.643-73.442-33.567Q-73.399-33.491-73.399-33.394Q-73.399-33.254-73.495-33.157Q-73.592-33.060-73.735-33.060Q-73.873-33.060-73.973-33.160Q-74.072-33.259-74.072-33.394Q-74.072-33.719-73.882-33.967Q-73.691-34.214-73.388-34.345Q-73.085-34.475-72.769-34.475Q-72.388-34.475-72.045-34.340Q-71.702-34.206-71.488-33.933Q-71.274-33.661-71.274-33.274Q-71.274-32.999-71.399-32.772Q-71.524-32.545-71.704-32.373Q-71.884-32.202-72.209-31.962Q-72.534-31.721-72.619-31.654L-73.375-31.050L-72.842-31.050Q-72.353-31.050-72.022-31.058Q-71.691-31.065-71.676-31.080Q-71.617-31.150-71.585-31.285Q-71.553-31.420-71.521-31.631L-71.274-31.631",[2846],[2830,6581,6582],{"transform":6570},[2835,6583],{"d":6558,"fill":2832,"stroke":2832,"className":6584,"style":4032},[2846],[2830,6586,6587],{"transform":6570},[2835,6588],{"d":6589,"fill":2832,"stroke":2832,"className":6590,"style":4032},"M-61.604-31.428Q-62.307-31.428-62.707-31.828Q-63.108-32.229-63.252-32.838Q-63.397-33.448-63.397-34.147Q-63.397-34.670-63.327-35.133Q-63.256-35.596-63.063-36.008Q-62.870-36.420-62.512-36.668Q-62.155-36.916-61.604-36.916Q-61.053-36.916-60.696-36.668Q-60.338-36.420-60.147-36.010Q-59.955-35.600-59.885-35.131Q-59.815-34.662-59.815-34.147Q-59.815-33.448-59.957-32.840Q-60.100-32.233-60.500-31.830Q-60.901-31.428-61.604-31.428M-61.604-31.686Q-61.131-31.686-60.899-32.121Q-60.666-32.557-60.612-33.096Q-60.557-33.635-60.557-34.276Q-60.557-35.272-60.741-35.965Q-60.924-36.659-61.604-36.659Q-61.971-36.659-62.192-36.420Q-62.413-36.182-62.508-35.825Q-62.604-35.467-62.629-35.096Q-62.655-34.725-62.655-34.276Q-62.655-33.635-62.600-33.096Q-62.545-32.557-62.313-32.121Q-62.080-31.686-61.604-31.686",[2846],[2835,6592],{"fill":2837,"d":6593,"style":2986},"M37.227-14.951h74.261",[2830,6595,6596,6602,6608,6613],{"stroke":2837},[2830,6597,6599],{"transform":6598},"translate(143.738 30.19)",[2835,6600],{"d":6545,"fill":2832,"stroke":2832,"className":6601,"style":4032},[2846],[2830,6603,6604],{"transform":6598},[2835,6605],{"d":6606,"fill":2832,"stroke":2832,"className":6607,"style":6553},"M-73.730-30.936Q-73.434-30.599-72.704-30.599Q-72.446-30.599-72.266-30.727Q-72.086-30.854-71.998-31.062Q-71.910-31.270-71.910-31.528Q-71.910-31.923-72.117-32.194Q-72.323-32.465-72.710-32.465L-73.176-32.465Q-73.240-32.480-73.255-32.542L-73.255-32.609Q-73.240-32.665-73.176-32.682L-72.774-32.706Q-72.564-32.706-72.395-32.848Q-72.227-32.990-72.134-33.204Q-72.042-33.418-72.042-33.634Q-72.042-33.922-72.227-34.087Q-72.411-34.253-72.704-34.253Q-72.965-34.253-73.189-34.185Q-73.413-34.118-73.560-33.960Q-73.431-33.942-73.352-33.853Q-73.273-33.763-73.273-33.634Q-73.273-33.497-73.368-33.402Q-73.463-33.306-73.604-33.306Q-73.738-33.306-73.835-33.403Q-73.932-33.500-73.932-33.634Q-73.932-33.922-73.741-34.113Q-73.551-34.305-73.270-34.390Q-72.988-34.475-72.704-34.475Q-72.429-34.475-72.128-34.384Q-71.828-34.294-71.620-34.105Q-71.412-33.916-71.412-33.634Q-71.412-33.265-71.658-32.993Q-71.904-32.720-72.276-32.591Q-71.857-32.498-71.540-32.215Q-71.222-31.932-71.222-31.534Q-71.222-31.171-71.441-30.905Q-71.661-30.640-72.007-30.500Q-72.353-30.359-72.704-30.359Q-72.927-30.359-73.174-30.407Q-73.422-30.456-73.642-30.566Q-73.861-30.675-73.993-30.854Q-74.125-31.033-74.125-31.288Q-74.125-31.437-74.023-31.540Q-73.920-31.642-73.771-31.642Q-73.621-31.642-73.519-31.540Q-73.416-31.437-73.416-31.288Q-73.416-31.156-73.505-31.055Q-73.595-30.954-73.730-30.936",[2846],[2830,6609,6610],{"transform":6598},[2835,6611],{"d":6558,"fill":2832,"stroke":2832,"className":6612,"style":4032},[2846],[2830,6614,6615],{"transform":6598},[2835,6616],{"d":6617,"fill":2832,"stroke":2832,"className":6618,"style":4032},"M-62.932-32.229Q-62.741-31.955-62.385-31.828Q-62.030-31.701-61.647-31.701Q-61.311-31.701-61.102-31.887Q-60.893-32.073-60.797-32.366Q-60.702-32.659-60.702-32.971Q-60.702-33.295-60.799-33.590Q-60.897-33.885-61.110-34.069Q-61.323-34.252-61.655-34.252L-62.221-34.252Q-62.252-34.252-62.282-34.282Q-62.311-34.311-62.311-34.338L-62.311-34.420Q-62.311-34.455-62.282-34.481Q-62.252-34.506-62.221-34.506L-61.741-34.541Q-61.455-34.541-61.258-34.746Q-61.061-34.951-60.965-35.246Q-60.870-35.541-60.870-35.819Q-60.870-36.198-61.069-36.436Q-61.268-36.674-61.647-36.674Q-61.967-36.674-62.256-36.567Q-62.545-36.459-62.709-36.237Q-62.530-36.237-62.407-36.110Q-62.284-35.983-62.284-35.811Q-62.284-35.639-62.409-35.514Q-62.534-35.389-62.709-35.389Q-62.881-35.389-63.006-35.514Q-63.131-35.639-63.131-35.811Q-63.131-36.178-62.907-36.426Q-62.682-36.674-62.342-36.795Q-62.002-36.916-61.647-36.916Q-61.299-36.916-60.936-36.795Q-60.573-36.674-60.325-36.424Q-60.077-36.174-60.077-35.819Q-60.077-35.334-60.395-34.951Q-60.713-34.569-61.190-34.397Q-60.639-34.287-60.239-33.901Q-59.838-33.514-59.838-32.979Q-59.838-32.522-60.102-32.166Q-60.366-31.811-60.788-31.619Q-61.209-31.428-61.647-31.428Q-62.057-31.428-62.450-31.563Q-62.842-31.698-63.108-31.983Q-63.373-32.268-63.373-32.686Q-63.373-32.881-63.241-33.010Q-63.108-33.139-62.916-33.139Q-62.791-33.139-62.688-33.080Q-62.584-33.022-62.522-32.916Q-62.459-32.811-62.459-32.686Q-62.459-32.491-62.594-32.360Q-62.729-32.229-62.932-32.229",[2846],[2830,6620,6621],{"fill":2953,"stroke":2953},[2830,6622,6623,6630,6636,6642,6648,6654,6660,6666,6672,6678,6684,6690,6696,6702,6708],{"fill":2953,"stroke":2837},[2830,6624,6626],{"transform":6625},"translate(34.265 -29.815)",[2835,6627],{"d":6628,"fill":2953,"stroke":2953,"className":6629,"style":4032},"M-79.007-33.323Q-79.007-33.819-78.757-34.244Q-78.507-34.670-78.087-34.916Q-77.667-35.162-77.167-35.162Q-76.628-35.162-76.237-35.037Q-75.847-34.912-75.847-34.498Q-75.847-34.393-75.897-34.301Q-75.948-34.209-76.040-34.159Q-76.132-34.108-76.241-34.108Q-76.347-34.108-76.438-34.159Q-76.530-34.209-76.581-34.301Q-76.632-34.393-76.632-34.498Q-76.632-34.721-76.464-34.826Q-76.686-34.885-77.159-34.885Q-77.456-34.885-77.671-34.746Q-77.886-34.608-78.017-34.377Q-78.147-34.147-78.206-33.877Q-78.265-33.608-78.265-33.323Q-78.265-32.928-78.132-32.578Q-77.999-32.229-77.727-32.012Q-77.456-31.795-77.058-31.795Q-76.683-31.795-76.407-32.012Q-76.132-32.229-76.030-32.588Q-76.015-32.651-75.952-32.651L-75.847-32.651Q-75.811-32.651-75.786-32.623Q-75.761-32.596-75.761-32.557L-75.761-32.534Q-75.893-32.053-76.278-31.785Q-76.663-31.518-77.167-31.518Q-77.530-31.518-77.864-31.655Q-78.198-31.791-78.458-32.041Q-78.718-32.291-78.862-32.627Q-79.007-32.963-79.007-33.323",[2846],[2830,6631,6632],{"transform":6625},[2835,6633],{"d":6634,"fill":2953,"stroke":2953,"className":6635,"style":4032},"M-73.578-31.596L-75.433-31.596L-75.433-31.893Q-75.160-31.893-74.992-31.940Q-74.824-31.987-74.824-32.155L-74.824-36.315Q-74.824-36.530-74.887-36.625Q-74.949-36.721-75.068-36.742Q-75.187-36.764-75.433-36.764L-75.433-37.061L-74.211-37.147L-74.211-34.444Q-74.086-34.655-73.898-34.805Q-73.711-34.955-73.484-35.039Q-73.258-35.123-73.012-35.123Q-71.844-35.123-71.844-34.045L-71.844-32.155Q-71.844-31.987-71.674-31.940Q-71.504-31.893-71.234-31.893L-71.234-31.596L-73.090-31.596L-73.090-31.893Q-72.816-31.893-72.648-31.940Q-72.480-31.987-72.480-32.155L-72.480-34.030Q-72.480-34.412-72.601-34.641Q-72.723-34.869-73.074-34.869Q-73.387-34.869-73.641-34.707Q-73.894-34.545-74.041-34.276Q-74.187-34.006-74.187-33.709L-74.187-32.155Q-74.187-31.987-74.017-31.940Q-73.848-31.893-73.578-31.893L-73.578-31.596M-70.789-33.291Q-70.789-33.795-70.533-34.227Q-70.277-34.659-69.842-34.910Q-69.406-35.162-68.906-35.162Q-68.519-35.162-68.178-35.018Q-67.836-34.873-67.574-34.612Q-67.312-34.350-67.170-34.014Q-67.027-33.678-67.027-33.291Q-67.027-32.799-67.291-32.389Q-67.555-31.979-67.984-31.748Q-68.414-31.518-68.906-31.518Q-69.398-31.518-69.832-31.750Q-70.266-31.983-70.527-32.391Q-70.789-32.799-70.789-33.291M-68.906-31.795Q-68.449-31.795-68.197-32.018Q-67.945-32.241-67.857-32.592Q-67.769-32.944-67.769-33.389Q-67.769-33.819-67.863-34.157Q-67.957-34.494-68.211-34.701Q-68.465-34.909-68.906-34.909Q-69.555-34.909-69.799-34.492Q-70.043-34.076-70.043-33.389Q-70.043-32.944-69.955-32.592Q-69.867-32.241-69.615-32.018Q-69.363-31.795-68.906-31.795",[2846],[2830,6637,6638],{"transform":6625},[2835,6639],{"d":6640,"fill":2953,"stroke":2953,"className":6641,"style":4032},"M-66.299-33.291Q-66.299-33.795-66.043-34.227Q-65.787-34.659-65.351-34.910Q-64.916-35.162-64.416-35.162Q-64.029-35.162-63.687-35.018Q-63.346-34.873-63.084-34.612Q-62.822-34.350-62.680-34.014Q-62.537-33.678-62.537-33.291Q-62.537-32.799-62.801-32.389Q-63.064-31.979-63.494-31.748Q-63.924-31.518-64.416-31.518Q-64.908-31.518-65.342-31.750Q-65.775-31.983-66.037-32.391Q-66.299-32.799-66.299-33.291M-64.416-31.795Q-63.959-31.795-63.707-32.018Q-63.455-32.241-63.367-32.592Q-63.279-32.944-63.279-33.389Q-63.279-33.819-63.373-34.157Q-63.467-34.494-63.721-34.701Q-63.975-34.909-64.416-34.909Q-65.064-34.909-65.308-34.492Q-65.553-34.076-65.553-33.389Q-65.553-32.944-65.465-32.592Q-65.377-32.241-65.125-32.018Q-64.873-31.795-64.416-31.795M-62.010-31.604L-62.010-32.826Q-62.010-32.854-61.978-32.885Q-61.947-32.916-61.924-32.916L-61.818-32.916Q-61.748-32.916-61.732-32.854Q-61.670-32.534-61.531-32.293Q-61.392-32.053-61.160-31.912Q-60.928-31.772-60.619-31.772Q-60.381-31.772-60.172-31.832Q-59.963-31.893-59.826-32.041Q-59.689-32.190-59.689-32.436Q-59.689-32.690-59.900-32.856Q-60.111-33.022-60.381-33.076L-61.002-33.190Q-61.408-33.268-61.709-33.524Q-62.010-33.780-62.010-34.155Q-62.010-34.522-61.808-34.744Q-61.607-34.967-61.283-35.065Q-60.959-35.162-60.619-35.162Q-60.154-35.162-59.857-34.955L-59.635-35.139Q-59.611-35.162-59.580-35.162L-59.529-35.162Q-59.498-35.162-59.471-35.135Q-59.443-35.108-59.443-35.076L-59.443-34.092Q-59.443-34.061-59.469-34.032Q-59.494-34.002-59.529-34.002L-59.635-34.002Q-59.670-34.002-59.697-34.030Q-59.725-34.057-59.725-34.092Q-59.725-34.491-59.976-34.711Q-60.228-34.932-60.627-34.932Q-60.982-34.932-61.266-34.809Q-61.549-34.686-61.549-34.381Q-61.549-34.162-61.348-34.030Q-61.146-33.897-60.900-33.854L-60.275-33.741Q-59.846-33.651-59.537-33.354Q-59.228-33.057-59.228-32.643Q-59.228-32.073-59.627-31.795Q-60.025-31.518-60.619-31.518Q-61.170-31.518-61.521-31.854L-61.818-31.541Q-61.842-31.518-61.877-31.518L-61.924-31.518Q-61.947-31.518-61.978-31.549Q-62.010-31.580-62.010-31.604M-58.701-33.350Q-58.701-33.830-58.469-34.246Q-58.236-34.662-57.826-34.912Q-57.416-35.162-56.939-35.162Q-56.209-35.162-55.810-34.721Q-55.412-34.280-55.412-33.549Q-55.412-33.444-55.506-33.420L-57.955-33.420L-57.955-33.350Q-57.955-32.940-57.834-32.584Q-57.713-32.229-57.441-32.012Q-57.170-31.795-56.740-31.795Q-56.377-31.795-56.080-32.024Q-55.783-32.252-55.682-32.604Q-55.674-32.651-55.588-32.666L-55.506-32.666Q-55.412-32.639-55.412-32.557Q-55.412-32.549-55.420-32.518Q-55.482-32.291-55.621-32.108Q-55.760-31.924-55.951-31.791Q-56.142-31.659-56.361-31.588Q-56.580-31.518-56.818-31.518Q-57.189-31.518-57.527-31.655Q-57.865-31.791-58.133-32.043Q-58.400-32.295-58.551-32.635Q-58.701-32.975-58.701-33.350M-57.947-33.659L-55.986-33.659Q-55.986-33.963-56.088-34.254Q-56.189-34.545-56.406-34.727Q-56.623-34.909-56.939-34.909Q-57.240-34.909-57.471-34.721Q-57.701-34.534-57.824-34.242Q-57.947-33.951-57.947-33.659",[2846],[2830,6643,6644],{"transform":6625},[2835,6645],{"d":6646,"fill":2953,"stroke":2953,"className":6647,"style":4032},"M-48.737-31.596L-51.897-31.596L-51.897-31.803Q-51.897-31.830-51.874-31.862L-50.522-33.260Q-50.143-33.647-49.895-33.936Q-49.647-34.225-49.473-34.582Q-49.300-34.940-49.300-35.330Q-49.300-35.678-49.432-35.971Q-49.565-36.264-49.819-36.442Q-50.073-36.619-50.428-36.619Q-50.788-36.619-51.079-36.424Q-51.370-36.229-51.514-35.901L-51.460-35.901Q-51.276-35.901-51.151-35.780Q-51.026-35.659-51.026-35.467Q-51.026-35.287-51.151-35.159Q-51.276-35.030-51.460-35.030Q-51.639-35.030-51.768-35.159Q-51.897-35.287-51.897-35.467Q-51.897-35.869-51.677-36.205Q-51.456-36.541-51.091-36.729Q-50.725-36.916-50.323-36.916Q-49.843-36.916-49.427-36.729Q-49.011-36.541-48.759-36.180Q-48.507-35.819-48.507-35.330Q-48.507-34.971-48.661-34.668Q-48.815-34.366-49.067-34.106Q-49.319-33.846-49.669-33.561Q-50.018-33.276-50.186-33.123L-51.116-32.284L-50.401-32.284Q-49.026-32.284-48.987-32.323Q-48.917-32.401-48.874-32.586Q-48.831-32.772-48.788-33.061L-48.507-33.061",[2846],[2830,6649,6650],{"transform":6625},[2835,6651],{"d":6652,"fill":2953,"stroke":2953,"className":6653,"style":4032},"M-45.002-33.291Q-45.002-33.795-44.746-34.227Q-44.490-34.659-44.054-34.910Q-43.619-35.162-43.119-35.162Q-42.732-35.162-42.390-35.018Q-42.049-34.873-41.787-34.612Q-41.525-34.350-41.383-34.014Q-41.240-33.678-41.240-33.291Q-41.240-32.799-41.504-32.389Q-41.767-31.979-42.197-31.748Q-42.627-31.518-43.119-31.518Q-43.611-31.518-44.045-31.750Q-44.478-31.983-44.740-32.391Q-45.002-32.799-45.002-33.291M-43.119-31.795Q-42.662-31.795-42.410-32.018Q-42.158-32.241-42.070-32.592Q-41.982-32.944-41.982-33.389Q-41.982-33.819-42.076-34.157Q-42.170-34.494-42.424-34.701Q-42.678-34.909-43.119-34.909Q-43.767-34.909-44.011-34.492Q-44.256-34.076-44.256-33.389Q-44.256-32.944-44.168-32.592Q-44.080-32.241-43.828-32.018Q-43.576-31.795-43.119-31.795M-38.689-31.596L-40.674-31.596L-40.674-31.893Q-40.400-31.893-40.232-31.940Q-40.064-31.987-40.064-32.155L-40.064-34.748L-40.705-34.748L-40.705-35.045L-40.064-35.045L-40.064-35.979Q-40.064-36.244-39.947-36.481Q-39.830-36.717-39.636-36.881Q-39.443-37.045-39.195-37.137Q-38.947-37.229-38.681-37.229Q-38.396-37.229-38.172-37.071Q-37.947-36.912-37.947-36.635Q-37.947-36.479-38.053-36.369Q-38.158-36.260-38.322-36.260Q-38.478-36.260-38.588-36.369Q-38.697-36.479-38.697-36.635Q-38.697-36.842-38.537-36.948Q-38.635-36.971-38.728-36.971Q-38.959-36.971-39.131-36.815Q-39.303-36.659-39.388-36.422Q-39.474-36.186-39.474-35.963L-39.474-35.045L-38.506-35.045L-38.506-34.748L-39.451-34.748L-39.451-32.155Q-39.451-31.987-39.224-31.940Q-38.998-31.893-38.689-31.893",[2846],[2830,6655,6656],{"transform":6625},[2835,6657],{"d":6658,"fill":2953,"stroke":2953,"className":6659,"style":4032},"M-34.063-31.819Q-34.063-32.244-33.979-32.694Q-33.895-33.143-33.739-33.561Q-33.582-33.979-33.368-34.366Q-33.153-34.752-32.879-35.108L-32.153-36.061L-33.063-36.061Q-34.559-36.061-34.598-36.022Q-34.668-35.940-34.715-35.750Q-34.762-35.561-34.805-35.284L-35.086-35.284L-34.817-37.002L-34.536-37.002L-34.536-36.979Q-34.536-36.834-34.018-36.791Q-33.500-36.748-33.008-36.748L-31.438-36.748L-31.438-36.557Q-31.446-36.518-31.461-36.491L-32.637-34.955Q-32.938-34.537-33.077-34.030Q-33.215-33.522-33.246-33.028Q-33.278-32.534-33.278-31.819Q-33.278-31.713-33.329-31.621Q-33.379-31.530-33.471-31.479Q-33.563-31.428-33.672-31.428Q-33.778-31.428-33.870-31.479Q-33.961-31.530-34.012-31.621Q-34.063-31.713-34.063-31.819",[2846],[2830,6661,6662],{"transform":6625},[2835,6663],{"d":6664,"fill":2953,"stroke":2953,"className":6665,"style":4032},"M-28.195-31.604L-28.195-32.826Q-28.195-32.854-28.163-32.885Q-28.132-32.916-28.109-32.916L-28.003-32.916Q-27.933-32.916-27.917-32.854Q-27.855-32.534-27.716-32.293Q-27.578-32.053-27.345-31.912Q-27.113-31.772-26.804-31.772Q-26.566-31.772-26.357-31.832Q-26.148-31.893-26.011-32.041Q-25.874-32.190-25.874-32.436Q-25.874-32.690-26.085-32.856Q-26.296-33.022-26.566-33.076L-27.187-33.190Q-27.593-33.268-27.894-33.524Q-28.195-33.780-28.195-34.155Q-28.195-34.522-27.994-34.744Q-27.792-34.967-27.468-35.065Q-27.144-35.162-26.804-35.162Q-26.339-35.162-26.042-34.955L-25.820-35.139Q-25.796-35.162-25.765-35.162L-25.714-35.162Q-25.683-35.162-25.656-35.135Q-25.628-35.108-25.628-35.076L-25.628-34.092Q-25.628-34.061-25.654-34.032Q-25.679-34.002-25.714-34.002L-25.820-34.002Q-25.855-34.002-25.882-34.030Q-25.910-34.057-25.910-34.092Q-25.910-34.491-26.162-34.711Q-26.413-34.932-26.812-34.932Q-27.167-34.932-27.451-34.809Q-27.734-34.686-27.734-34.381Q-27.734-34.162-27.533-34.030Q-27.331-33.897-27.085-33.854L-26.460-33.741Q-26.031-33.651-25.722-33.354Q-25.413-33.057-25.413-32.643Q-25.413-32.073-25.812-31.795Q-26.210-31.518-26.804-31.518Q-27.355-31.518-27.706-31.854L-28.003-31.541Q-28.027-31.518-28.062-31.518L-28.109-31.518Q-28.132-31.518-28.163-31.549Q-28.195-31.580-28.195-31.604M-22.972-31.596L-24.804-31.596L-24.804-31.893Q-24.531-31.893-24.363-31.940Q-24.195-31.987-24.195-32.155L-24.195-36.315Q-24.195-36.530-24.257-36.625Q-24.320-36.721-24.439-36.742Q-24.558-36.764-24.804-36.764L-24.804-37.061L-23.581-37.147L-23.581-32.155Q-23.581-31.987-23.413-31.940Q-23.246-31.893-22.972-31.893L-22.972-31.596M-22.527-33.291Q-22.527-33.795-22.271-34.227Q-22.015-34.659-21.580-34.910Q-21.144-35.162-20.644-35.162Q-20.257-35.162-19.915-35.018Q-19.574-34.873-19.312-34.612Q-19.050-34.350-18.908-34.014Q-18.765-33.678-18.765-33.291Q-18.765-32.799-19.029-32.389Q-19.292-31.979-19.722-31.748Q-20.152-31.518-20.644-31.518Q-21.136-31.518-21.570-31.750Q-22.003-31.983-22.265-32.391Q-22.527-32.799-22.527-33.291M-20.644-31.795Q-20.187-31.795-19.935-32.018Q-19.683-32.241-19.595-32.592Q-19.507-32.944-19.507-33.389Q-19.507-33.819-19.601-34.157Q-19.695-34.494-19.949-34.701Q-20.203-34.909-20.644-34.909Q-21.292-34.909-21.537-34.492Q-21.781-34.076-21.781-33.389Q-21.781-32.944-21.693-32.592Q-21.605-32.241-21.353-32.018Q-21.101-31.795-20.644-31.795M-17.656-32.557L-17.656-34.748L-18.359-34.748L-18.359-35.002Q-18.003-35.002-17.761-35.235Q-17.519-35.467-17.408-35.815Q-17.296-36.162-17.296-36.518L-17.015-36.518L-17.015-35.045L-15.839-35.045L-15.839-34.748L-17.015-34.748L-17.015-32.573Q-17.015-32.252-16.896-32.024Q-16.777-31.795-16.496-31.795Q-16.316-31.795-16.199-31.918Q-16.081-32.041-16.029-32.221Q-15.976-32.401-15.976-32.573L-15.976-33.045L-15.695-33.045L-15.695-32.557Q-15.695-32.303-15.800-32.063Q-15.906-31.823-16.103-31.670Q-16.300-31.518-16.558-31.518Q-16.874-31.518-17.126-31.641Q-17.378-31.764-17.517-31.998Q-17.656-32.233-17.656-32.557M-14.933-31.604L-14.933-32.826Q-14.933-32.854-14.902-32.885Q-14.871-32.916-14.847-32.916L-14.742-32.916Q-14.671-32.916-14.656-32.854Q-14.593-32.534-14.455-32.293Q-14.316-32.053-14.083-31.912Q-13.851-31.772-13.542-31.772Q-13.304-31.772-13.095-31.832Q-12.886-31.893-12.749-32.041Q-12.613-32.190-12.613-32.436Q-12.613-32.690-12.824-32.856Q-13.035-33.022-13.304-33.076L-13.925-33.190Q-14.331-33.268-14.632-33.524Q-14.933-33.780-14.933-34.155Q-14.933-34.522-14.732-34.744Q-14.531-34.967-14.206-35.065Q-13.882-35.162-13.542-35.162Q-13.078-35.162-12.781-34.955L-12.558-35.139Q-12.535-35.162-12.503-35.162L-12.453-35.162Q-12.421-35.162-12.394-35.135Q-12.367-35.108-12.367-35.076L-12.367-34.092Q-12.367-34.061-12.392-34.032Q-12.417-34.002-12.453-34.002L-12.558-34.002Q-12.593-34.002-12.621-34.030Q-12.648-34.057-12.648-34.092Q-12.648-34.491-12.900-34.711Q-13.152-34.932-13.550-34.932Q-13.906-34.932-14.189-34.809Q-14.472-34.686-14.472-34.381Q-14.472-34.162-14.271-34.030Q-14.070-33.897-13.824-33.854L-13.199-33.741Q-12.769-33.651-12.460-33.354Q-12.152-33.057-12.152-32.643Q-12.152-32.073-12.550-31.795Q-12.949-31.518-13.542-31.518Q-14.093-31.518-14.445-31.854L-14.742-31.541Q-14.765-31.518-14.800-31.518L-14.847-31.518Q-14.871-31.518-14.902-31.549Q-14.933-31.580-14.933-31.604",[2846],[2830,6667,6668],{"transform":6625},[2835,6669],{"d":6670,"fill":2953,"stroke":2953,"className":6671,"style":4032},"M-6.715-31.596L-8.700-31.596L-8.700-31.893Q-8.426-31.893-8.258-31.940Q-8.090-31.987-8.090-32.155L-8.090-34.748L-8.731-34.748L-8.731-35.045L-8.090-35.045L-8.090-35.979Q-8.090-36.244-7.973-36.481Q-7.856-36.717-7.663-36.881Q-7.469-37.045-7.221-37.137Q-6.973-37.229-6.707-37.229Q-6.422-37.229-6.198-37.071Q-5.973-36.912-5.973-36.635Q-5.973-36.479-6.079-36.369Q-6.184-36.260-6.348-36.260Q-6.504-36.260-6.614-36.369Q-6.723-36.479-6.723-36.635Q-6.723-36.842-6.563-36.948Q-6.661-36.971-6.754-36.971Q-6.985-36.971-7.157-36.815Q-7.329-36.659-7.415-36.422Q-7.500-36.186-7.500-35.963L-7.500-35.045L-6.532-35.045L-6.532-34.748L-7.477-34.748L-7.477-32.155Q-7.477-31.987-7.250-31.940Q-7.024-31.893-6.715-31.893L-6.715-31.596M-6.188-33.291Q-6.188-33.795-5.932-34.227Q-5.676-34.659-5.241-34.910Q-4.805-35.162-4.305-35.162Q-3.918-35.162-3.577-35.018Q-3.235-34.873-2.973-34.612Q-2.711-34.350-2.569-34.014Q-2.426-33.678-2.426-33.291Q-2.426-32.799-2.690-32.389Q-2.954-31.979-3.383-31.748Q-3.813-31.518-4.305-31.518Q-4.797-31.518-5.231-31.750Q-5.665-31.983-5.926-32.391Q-6.188-32.799-6.188-33.291M-4.305-31.795Q-3.848-31.795-3.596-32.018Q-3.344-32.241-3.256-32.592Q-3.168-32.944-3.168-33.389Q-3.168-33.819-3.262-34.157Q-3.356-34.494-3.610-34.701Q-3.864-34.909-4.305-34.909Q-4.954-34.909-5.198-34.492Q-5.442-34.076-5.442-33.389Q-5.442-32.944-5.354-32.592Q-5.266-32.241-5.014-32.018Q-4.762-31.795-4.305-31.795M0.066-31.596L-1.915-31.596L-1.915-31.893Q-1.645-31.893-1.477-31.938Q-1.309-31.983-1.309-32.155L-1.309-34.291Q-1.309-34.506-1.372-34.602Q-1.434-34.698-1.551-34.719Q-1.668-34.741-1.915-34.741L-1.915-35.037L-0.747-35.123L-0.747-34.338Q-0.668-34.549-0.516-34.735Q-0.364-34.920-0.165-35.022Q0.035-35.123 0.261-35.123Q0.507-35.123 0.699-34.979Q0.890-34.834 0.890-34.604Q0.890-34.448 0.785-34.338Q0.679-34.229 0.523-34.229Q0.367-34.229 0.257-34.338Q0.148-34.448 0.148-34.604Q0.148-34.764 0.253-34.869Q-0.071-34.869-0.286-34.641Q-0.500-34.412-0.596-34.073Q-0.692-33.733-0.692-33.428L-0.692-32.155Q-0.692-31.987-0.465-31.940Q-0.239-31.893 0.066-31.893",[2846],[2830,6673,6674],{"transform":6625},[2835,6675],{"d":6676,"fill":2953,"stroke":2953,"className":6677,"style":4032},"M5.125-31.596L4.844-31.596L4.844-36.315Q4.844-36.530 4.782-36.625Q4.719-36.721 4.602-36.742Q4.485-36.764 4.239-36.764L4.239-37.061L5.461-37.147L5.461-34.659Q5.938-35.123 6.637-35.123Q7.118-35.123 7.526-34.879Q7.934-34.635 8.170-34.221Q8.407-33.807 8.407-33.323Q8.407-32.948 8.258-32.619Q8.110-32.291 7.840-32.039Q7.571-31.787 7.227-31.653Q6.883-31.518 6.524-31.518Q6.203-31.518 5.905-31.666Q5.606-31.815 5.399-32.076L5.125-31.596M5.485-34.268L5.485-32.428Q5.637-32.131 5.897-31.951Q6.157-31.772 6.469-31.772Q6.895-31.772 7.162-31.991Q7.430-32.209 7.545-32.555Q7.660-32.901 7.660-33.323Q7.660-33.971 7.412-34.420Q7.164-34.869 6.567-34.869Q6.231-34.869 5.942-34.711Q5.653-34.553 5.485-34.268M9.028-32.428Q9.028-32.912 9.430-33.207Q9.832-33.502 10.383-33.621Q10.934-33.741 11.426-33.741L11.426-34.030Q11.426-34.256 11.311-34.463Q11.196-34.670 10.998-34.789Q10.801-34.909 10.571-34.909Q10.145-34.909 9.860-34.803Q9.930-34.776 9.977-34.721Q10.024-34.666 10.049-34.596Q10.075-34.526 10.075-34.451Q10.075-34.346 10.024-34.254Q9.973-34.162 9.881-34.112Q9.789-34.061 9.684-34.061Q9.578-34.061 9.487-34.112Q9.395-34.162 9.344-34.254Q9.293-34.346 9.293-34.451Q9.293-34.869 9.682-35.016Q10.071-35.162 10.571-35.162Q10.903-35.162 11.256-35.032Q11.610-34.901 11.838-34.647Q12.067-34.393 12.067-34.045L12.067-32.244Q12.067-32.112 12.139-32.002Q12.211-31.893 12.340-31.893Q12.465-31.893 12.534-31.998Q12.602-32.104 12.602-32.244L12.602-32.756L12.883-32.756L12.883-32.244Q12.883-32.041 12.766-31.883Q12.649-31.725 12.467-31.641Q12.285-31.557 12.082-31.557Q11.852-31.557 11.700-31.729Q11.547-31.901 11.516-32.131Q11.356-31.850 11.047-31.684Q10.739-31.518 10.387-31.518Q9.875-31.518 9.452-31.741Q9.028-31.963 9.028-32.428M9.715-32.428Q9.715-32.143 9.942-31.957Q10.168-31.772 10.461-31.772Q10.707-31.772 10.932-31.889Q11.157-32.006 11.291-32.209Q11.426-32.412 11.426-32.666L11.426-33.498Q11.160-33.498 10.875-33.444Q10.590-33.389 10.319-33.260Q10.047-33.131 9.881-32.924Q9.715-32.717 9.715-32.428M15.184-31.596L13.203-31.596L13.203-31.893Q13.473-31.893 13.641-31.938Q13.809-31.983 13.809-32.155L13.809-34.291Q13.809-34.506 13.746-34.602Q13.684-34.698 13.567-34.719Q13.450-34.741 13.203-34.741L13.203-35.037L14.371-35.123L14.371-34.338Q14.450-34.549 14.602-34.735Q14.754-34.920 14.953-35.022Q15.153-35.123 15.379-35.123Q15.625-35.123 15.817-34.979Q16.008-34.834 16.008-34.604Q16.008-34.448 15.903-34.338Q15.797-34.229 15.641-34.229Q15.485-34.229 15.375-34.338Q15.266-34.448 15.266-34.604Q15.266-34.764 15.371-34.869Q15.047-34.869 14.832-34.641Q14.618-34.412 14.522-34.073Q14.426-33.733 14.426-33.428L14.426-32.155Q14.426-31.987 14.653-31.940Q14.879-31.893 15.184-31.893L15.184-31.596M16.532-31.604L16.532-32.826Q16.532-32.854 16.563-32.885Q16.594-32.916 16.618-32.916L16.723-32.916Q16.793-32.916 16.809-32.854Q16.871-32.534 17.010-32.293Q17.149-32.053 17.381-31.912Q17.614-31.772 17.922-31.772Q18.160-31.772 18.369-31.832Q18.578-31.893 18.715-32.041Q18.852-32.190 18.852-32.436Q18.852-32.690 18.641-32.856Q18.430-33.022 18.160-33.076L17.539-33.190Q17.133-33.268 16.832-33.524Q16.532-33.780 16.532-34.155Q16.532-34.522 16.733-34.744Q16.934-34.967 17.258-35.065Q17.582-35.162 17.922-35.162Q18.387-35.162 18.684-34.955L18.907-35.139Q18.930-35.162 18.961-35.162L19.012-35.162Q19.043-35.162 19.071-35.135Q19.098-35.108 19.098-35.076L19.098-34.092Q19.098-34.061 19.073-34.032Q19.047-34.002 19.012-34.002L18.907-34.002Q18.871-34.002 18.844-34.030Q18.817-34.057 18.817-34.092Q18.817-34.491 18.565-34.711Q18.313-34.932 17.914-34.932Q17.559-34.932 17.276-34.809Q16.993-34.686 16.993-34.381Q16.993-34.162 17.194-34.030Q17.395-33.897 17.641-33.854L18.266-33.741Q18.696-33.651 19.004-33.354Q19.313-33.057 19.313-32.643Q19.313-32.073 18.914-31.795Q18.516-31.518 17.922-31.518Q17.371-31.518 17.020-31.854L16.723-31.541Q16.700-31.518 16.664-31.518L16.618-31.518Q16.594-31.518 16.563-31.549Q16.532-31.580 16.532-31.604M20.321-32.061Q20.321-32.244 20.457-32.381Q20.594-32.518 20.785-32.518Q20.977-32.518 21.110-32.385Q21.243-32.252 21.243-32.061Q21.243-31.862 21.110-31.729Q20.977-31.596 20.785-31.596Q20.594-31.596 20.457-31.733Q20.321-31.869 20.321-32.061M20.321-34.588Q20.321-34.772 20.457-34.909Q20.594-35.045 20.785-35.045Q20.977-35.045 21.110-34.912Q21.243-34.780 21.243-34.588Q21.243-34.389 21.110-34.256Q20.977-34.123 20.785-34.123Q20.594-34.123 20.457-34.260Q20.321-34.397 20.321-34.588",[2846],[2830,6679,6680],{"transform":6625},[2835,6681],{"d":6682,"fill":2953,"stroke":2953,"className":6683,"style":4032},"M29.063-28.829Q28.551-29.239 28.162-29.788Q27.773-30.337 27.527-30.966Q27.281-31.595 27.168-32.251Q27.055-32.907 27.055-33.595Q27.055-34.290 27.168-34.948Q27.281-35.606 27.527-36.237Q27.773-36.868 28.162-37.417Q28.551-37.966 29.063-38.372Q29.086-38.395 29.117-38.395L29.223-38.395Q29.258-38.395 29.283-38.368Q29.309-38.341 29.309-38.302Q29.309-38.259 29.285-38.235Q28.844-37.802 28.535-37.265Q28.227-36.727 28.041-36.128Q27.855-35.528 27.775-34.888Q27.695-34.247 27.695-33.595Q27.695-32.946 27.777-32.307Q27.859-31.669 28.041-31.077Q28.223-30.485 28.531-29.948Q28.840-29.411 29.285-28.966Q29.309-28.938 29.309-28.899Q29.309-28.860 29.285-28.833Q29.262-28.806 29.223-28.806L29.117-28.806Q29.086-28.806 29.063-28.829",[2846],[2830,6685,6686],{"transform":6625},[2835,6687],{"d":6688,"fill":2953,"stroke":2953,"className":6689,"style":6553},"M30.985-35.434Q30.985-35.873 31.101-36.308Q31.216-36.743 31.432-37.145Q31.647-37.546 31.937-37.889L32.502-38.554L31.793-38.554Q30.557-38.554 30.528-38.525Q30.434-38.419 30.367-37.974L30.121-37.974L30.337-39.310L30.581-39.310L30.581-39.292Q30.581-39.231 30.654-39.193Q30.727-39.154 30.792-39.154Q31.210-39.119 31.843-39.119L33.126-39.119L33.126-38.952Q33.126-38.932 33.109-38.897L32.154-37.769Q31.791-37.344 31.691-36.780Q31.591-36.216 31.591-35.422Q31.591-35.299 31.500-35.214Q31.410-35.129 31.290-35.129Q31.161-35.129 31.073-35.217Q30.985-35.305 30.985-35.434",[2846],[2830,6691,6692],{"transform":6625},[2835,6693],{"d":6694,"fill":2953,"stroke":2953,"className":6695,"style":6553},"M32.687-28.459L30.077-28.459L30.077-28.644Q30.083-28.667 30.103-28.693L31.254-29.748Q31.594-30.059 31.774-30.245Q31.955-30.431 32.100-30.691Q32.245-30.952 32.245-31.248Q32.245-31.521 32.119-31.736Q31.993-31.951 31.773-32.071Q31.553-32.191 31.278-32.191Q31.102-32.191 30.932-32.134Q30.762-32.077 30.630-31.970Q30.499-31.863 30.419-31.705Q30.507-31.705 30.585-31.661Q30.663-31.617 30.707-31.541Q30.750-31.465 30.750-31.368Q30.750-31.228 30.654-31.131Q30.557-31.034 30.414-31.034Q30.276-31.034 30.176-31.134Q30.077-31.233 30.077-31.368Q30.077-31.693 30.267-31.941Q30.458-32.188 30.761-32.319Q31.064-32.449 31.380-32.449Q31.761-32.449 32.104-32.314Q32.447-32.180 32.661-31.907Q32.875-31.635 32.875-31.248Q32.875-30.973 32.750-30.746Q32.625-30.519 32.445-30.347Q32.265-30.176 31.940-29.936Q31.615-29.695 31.530-29.628L30.774-29.024L31.307-29.024Q31.796-29.024 32.127-29.032Q32.459-29.039 32.473-29.054Q32.532-29.124 32.564-29.259Q32.596-29.394 32.628-29.605L32.875-29.605",[2846],[2830,6697,6698],{"transform":6625},[2835,6699],{"d":6700,"fill":2953,"stroke":2953,"className":6701,"style":4032},"M33.833-28.806L33.727-28.806Q33.641-28.806 33.641-28.899Q33.641-28.934 33.665-28.966Q34.114-29.407 34.428-29.958Q34.743-30.509 34.918-31.089Q35.094-31.669 35.176-32.307Q35.258-32.946 35.258-33.595Q35.258-34.247 35.178-34.888Q35.098-35.528 34.918-36.116Q34.739-36.704 34.422-37.255Q34.106-37.806 33.665-38.235Q33.641-38.259 33.641-38.302Q33.641-38.395 33.727-38.395L33.833-38.395Q33.864-38.395 33.887-38.372Q34.407-37.966 34.795-37.417Q35.184-36.868 35.428-36.241Q35.672-35.614 35.786-34.952Q35.899-34.290 35.899-33.595Q35.899-32.669 35.690-31.794Q35.481-30.919 35.030-30.149Q34.579-29.380 33.887-28.829Q33.864-28.806 33.833-28.806",[2846],[2830,6703,6704],{"transform":6625},[2835,6705],{"d":6706,"fill":2953,"stroke":2953,"className":6707,"style":4032},"M45.531-32.573L40.218-32.573Q40.140-32.580 40.091-32.629Q40.043-32.678 40.043-32.756Q40.043-32.826 40.090-32.877Q40.136-32.928 40.218-32.940L45.531-32.940Q45.605-32.928 45.652-32.877Q45.699-32.826 45.699-32.756Q45.699-32.678 45.650-32.629Q45.601-32.580 45.531-32.573M45.531-34.260L40.218-34.260Q40.140-34.268 40.091-34.317Q40.043-34.366 40.043-34.444Q40.043-34.514 40.090-34.565Q40.136-34.616 40.218-34.627L45.531-34.627Q45.605-34.616 45.652-34.565Q45.699-34.514 45.699-34.444Q45.699-34.366 45.650-34.317Q45.601-34.268 45.531-34.260",[2846],[2830,6709,6710],{"transform":6625},[2835,6711],{"d":6712,"fill":2953,"stroke":2953,"className":6713,"style":4032},"M52.128-31.596L48.968-31.596L48.968-31.803Q48.968-31.830 48.991-31.862L50.343-33.260Q50.722-33.647 50.970-33.936Q51.218-34.225 51.392-34.582Q51.565-34.940 51.565-35.330Q51.565-35.678 51.433-35.971Q51.300-36.264 51.046-36.442Q50.792-36.619 50.437-36.619Q50.077-36.619 49.786-36.424Q49.495-36.229 49.351-35.901L49.405-35.901Q49.589-35.901 49.714-35.780Q49.839-35.659 49.839-35.467Q49.839-35.287 49.714-35.159Q49.589-35.030 49.405-35.030Q49.226-35.030 49.097-35.159Q48.968-35.287 48.968-35.467Q48.968-35.869 49.188-36.205Q49.409-36.541 49.774-36.729Q50.140-36.916 50.542-36.916Q51.022-36.916 51.438-36.729Q51.855-36.541 52.106-36.180Q52.358-35.819 52.358-35.330Q52.358-34.971 52.204-34.668Q52.050-34.366 51.798-34.106Q51.546-33.846 51.196-33.561Q50.847-33.276 50.679-33.123L49.749-32.284L50.464-32.284Q51.839-32.284 51.878-32.323Q51.948-32.401 51.991-32.586Q52.034-32.772 52.077-33.061L52.358-33.061L52.128-31.596M56.382-31.596L53.589-31.596L53.589-31.893Q54.651-31.893 54.651-32.155L54.651-36.323Q54.222-36.108 53.542-36.108L53.542-36.405Q54.562-36.405 55.077-36.916L55.222-36.916Q55.296-36.897 55.315-36.819L55.315-32.155Q55.315-31.893 56.382-31.893",[2846],[3003,6715,6717,6718,5926,6733,6748,6749,6764,6765],{"className":6716},[3006],"Stars and bars: ",[394,6719,6721],{"className":6720},[397],[394,6722,6724],{"className":6723,"ariaHidden":402},[401],[394,6725,6727,6730],{"className":6726},[406],[394,6728],{"className":6729,"style":639},[410],[394,6731,5573],{"className":6732},[415],[394,6734,6736],{"className":6735},[397],[394,6737,6739],{"className":6738,"ariaHidden":402},[401],[394,6740,6742,6745],{"className":6741},[406],[394,6743],{"className":6744,"style":639},[410],[394,6746,520],{"className":6747},[415]," bars in ",[394,6750,6752],{"className":6751},[397],[394,6753,6755],{"className":6754,"ariaHidden":402},[401],[394,6756,6758,6761],{"className":6757},[406],[394,6759],{"className":6760,"style":639},[410],[394,6762,6353],{"className":6763},[415]," slots encode ",[394,6766,6768],{"className":6767},[397],[394,6769,6771,6924],{"className":6770,"ariaHidden":402},[401],[394,6772,6774,6777,6780,6820,6823,6826,6866,6869,6872,6912,6915,6918,6921],{"className":6773},[406],[394,6775],{"className":6776,"style":535},[410],[394,6778,540],{"className":6779},[539],[394,6781,6783,6786],{"className":6782},[415],[394,6784,4243],{"className":6785},[415,419],[394,6787,6789],{"className":6788},[423],[394,6790,6792,6812],{"className":6791},[427,913],[394,6793,6795,6809],{"className":6794},[431],[394,6796,6798],{"className":6797,"style":5013},[435],[394,6799,6800,6803],{"style":5016},[394,6801],{"className":6802,"style":443},[442],[394,6804,6806],{"className":6805},[447,448,449,450],[394,6807,461],{"className":6808},[415,450],[394,6810,983],{"className":6811},[982],[394,6813,6815],{"className":6814},[431],[394,6816,6818],{"className":6817,"style":5035},[435],[394,6819],{},[394,6821,769],{"className":6822},[768],[394,6824],{"className":6825,"style":734},[465],[394,6827,6829,6832],{"className":6828},[415],[394,6830,4243],{"className":6831},[415,419],[394,6833,6835],{"className":6834},[423],[394,6836,6838,6858],{"className":6837},[427,913],[394,6839,6841,6855],{"className":6840},[431],[394,6842,6844],{"className":6843,"style":5013},[435],[394,6845,6846,6849],{"style":5016},[394,6847],{"className":6848,"style":443},[442],[394,6850,6852],{"className":6851},[447,448,449,450],[394,6853,520],{"className":6854},[415,450],[394,6856,983],{"className":6857},[982],[394,6859,6861],{"className":6860},[431],[394,6862,6864],{"className":6863,"style":5035},[435],[394,6865],{},[394,6867,769],{"className":6868},[768],[394,6870],{"className":6871,"style":734},[465],[394,6873,6875,6878],{"className":6874},[415],[394,6876,4243],{"className":6877},[415,419],[394,6879,6881],{"className":6880},[423],[394,6882,6884,6904],{"className":6883},[427,913],[394,6885,6887,6901],{"className":6886},[431],[394,6888,6890],{"className":6889,"style":5013},[435],[394,6891,6892,6895],{"style":5016},[394,6893],{"className":6894,"style":443},[442],[394,6896,6898],{"className":6897},[447,448,449,450],[394,6899,3386],{"className":6900},[415,450],[394,6902,983],{"className":6903},[982],[394,6905,6907],{"className":6906},[431],[394,6908,6910],{"className":6909,"style":5035},[435],[394,6911],{},[394,6913,563],{"className":6914},[562],[394,6916],{"className":6917,"style":466},[465],[394,6919,674],{"className":6920},[470],[394,6922],{"className":6923,"style":466},[465],[394,6925,6927,6930,6933,6936,6939,6942,6945,6948,6951,6954],{"className":6926},[406],[394,6928],{"className":6929,"style":535},[410],[394,6931,540],{"className":6932},[539],[394,6934,520],{"className":6935},[415],[394,6937,769],{"className":6938},[768],[394,6940],{"className":6941,"style":734},[465],[394,6943,780],{"className":6944},[415],[394,6946,769],{"className":6947},[768],[394,6949],{"className":6950,"style":734},[465],[394,6952,3386],{"className":6953},[415],[394,6955,563],{"className":6956},[562],[578,6958,6960,6961],{"id":6959},"computing-knmodp","Computing ",[394,6962,6964],{"className":6963},[397],[394,6965,6967,7061],{"className":6966,"ariaHidden":402},[401],[394,6968,6970,6973,7039,7043,7046,7055,7058],{"className":6969},[406],[394,6971],{"className":6972,"style":1497},[410],[394,6974,6976,6982,7033],{"className":6975},[415],[394,6977,6979],{"className":6978,"style":1226},[539,1225],[394,6980,540],{"className":6981},[1230,1507],[394,6983,6985],{"className":6984},[909],[394,6986,6988,7025],{"className":6987},[427,913],[394,6989,6991,7022],{"className":6990},[431],[394,6992,6994,7008],{"className":6993,"style":1520},[435],[394,6995,6996,6999],{"style":1523},[394,6997],{"className":6998,"style":443},[442],[394,7000,7002],{"className":7001},[447,448,449,450],[394,7003,7005],{"className":7004},[415,450],[394,7006,2068],{"className":7007,"style":2067},[415,419,450],[394,7009,7010,7013],{"style":1538},[394,7011],{"className":7012,"style":443},[442],[394,7014,7016],{"className":7015},[447,448,449,450],[394,7017,7019],{"className":7018},[415,450],[394,7020,605],{"className":7021},[415,419,450],[394,7023,983],{"className":7024},[982],[394,7026,7028],{"className":7027},[431],[394,7029,7031],{"className":7030,"style":1560},[435],[394,7032],{},[394,7034,7036],{"className":7035,"style":1226},[562,1225],[394,7037,563],{"className":7038},[1230,1507],[394,7040],{"className":7041,"style":7042},[465],"margin-right:0.0556em;",[394,7044],{"className":7045,"style":626},[465],[394,7047,7049],{"className":7048},[516],[394,7050,7052],{"className":7051},[415],[394,7053,551],{"className":7054},[415,550],[394,7056],{"className":7057,"style":7042},[465],[394,7059],{"className":7060,"style":626},[465],[394,7062,7064,7067],{"className":7063},[406],[394,7065],{"className":7066,"style":4597},[410],[394,7068,381],{"className":7069},[415,419],[381,7071,7072,7073,7088,7089,7153,7154,7202,7203,7221,7222,7224,7225,7240,7241,7374,7375,7445],{},"Competitive and large-scale problems ask for counts modulo a\nprime ",[394,7074,7076],{"className":7075},[397],[394,7077,7079],{"className":7078,"ariaHidden":402},[401],[394,7080,7082,7085],{"className":7081},[406],[394,7083],{"className":7084,"style":4597},[410],[394,7086,381],{"className":7087},[415,419]," (typically ",[394,7090,7092],{"className":7091},[397],[394,7093,7095,7144],{"className":7094,"ariaHidden":402},[401],[394,7096,7098,7102,7105,7135,7138,7141],{"className":7097},[406],[394,7099],{"className":7100,"style":7101},[410],"height:0.8974em;vertical-align:-0.0833em;",[394,7103,461],{"className":7104},[415],[394,7106,7108,7111],{"className":7107},[415],[394,7109,780],{"className":7110},[415],[394,7112,7114],{"className":7113},[423],[394,7115,7117],{"className":7116},[427],[394,7118,7120],{"className":7119},[431],[394,7121,7123],{"className":7122,"style":411},[435],[394,7124,7125,7128],{"style":438},[394,7126],{"className":7127,"style":443},[442],[394,7129,7131],{"className":7130},[447,448,449,450],[394,7132,7134],{"className":7133},[415,450],"9",[394,7136],{"className":7137,"style":626},[465],[394,7139,1080],{"className":7140},[516],[394,7142],{"className":7143,"style":626},[465],[394,7145,7147,7150],{"className":7146},[406],[394,7148],{"className":7149,"style":639},[410],[394,7151,6353],{"className":7152},[415],") because the true values are astronomically large.\nThe factorial formula has a division by ",[394,7155,7157],{"className":7156},[397],[394,7158,7160,7190],{"className":7159,"ariaHidden":402},[401],[394,7161,7163,7166,7169,7172,7175,7178,7181,7184,7187],{"className":7162},[406],[394,7164],{"className":7165,"style":535},[410],[394,7167,822],{"className":7168,"style":821},[415,419],[394,7170,667],{"className":7171},[562],[394,7173],{"className":7174,"style":734},[465],[394,7176,540],{"className":7177},[539],[394,7179,605],{"className":7180},[415,419],[394,7182],{"className":7183,"style":626},[465],[394,7185,457],{"className":7186},[516],[394,7188],{"className":7189,"style":626},[465],[394,7191,7193,7196,7199],{"className":7192},[406],[394,7194],{"className":7195,"style":535},[410],[394,7197,822],{"className":7198,"style":821},[415,419],[394,7200,952],{"className":7201},[562],", and ",[389,7204,7205,7206],{},"division is not defined\nmodulo ",[394,7207,7209],{"className":7208},[397],[394,7210,7212],{"className":7211,"ariaHidden":402},[401],[394,7213,7215,7218],{"className":7214},[406],[394,7216],{"className":7217,"style":4597},[410],[394,7219,381],{"className":7220},[415,419],"; what stands in for it is multiplication by a ",[384,7223,391],{"href":312},". Since ",[394,7226,7228],{"className":7227},[397],[394,7229,7231],{"className":7230,"ariaHidden":402},[401],[394,7232,7234,7237],{"className":7233},[406],[394,7235],{"className":7236,"style":4597},[410],[394,7238,381],{"className":7239},[415,419]," is prime,\nFermat gives ",[394,7242,7244],{"className":7243},[397],[394,7245,7247,7297,7347],{"className":7246,"ariaHidden":402},[401],[394,7248,7250,7253,7288,7291,7294],{"className":7249},[406],[394,7251],{"className":7252,"style":411},[410],[394,7254,7256,7259],{"className":7255},[415],[394,7257,384],{"className":7258},[415,419],[394,7260,7262],{"className":7261},[423],[394,7263,7265],{"className":7264},[427],[394,7266,7268],{"className":7267},[431],[394,7269,7271],{"className":7270,"style":411},[435],[394,7272,7273,7276],{"style":438},[394,7274],{"className":7275,"style":443},[442],[394,7277,7279],{"className":7278},[447,448,449,450],[394,7280,7282,7285],{"className":7281},[415,450],[394,7283,457],{"className":7284},[415,450],[394,7286,461],{"className":7287},[415,450],[394,7289],{"className":7290,"style":466},[465],[394,7292,471],{"className":7293},[470],[394,7295],{"className":7296,"style":466},[465],[394,7298,7300,7303,7341,7344],{"className":7299},[406],[394,7301],{"className":7302,"style":411},[410],[394,7304,7306,7309],{"className":7305},[415],[394,7307,384],{"className":7308},[415,419],[394,7310,7312],{"className":7311},[423],[394,7313,7315],{"className":7314},[427],[394,7316,7318],{"className":7317},[431],[394,7319,7321],{"className":7320,"style":411},[435],[394,7322,7323,7326],{"style":438},[394,7324],{"className":7325,"style":443},[442],[394,7327,7329],{"className":7328},[447,448,449,450],[394,7330,7332,7335,7338],{"className":7331},[415,450],[394,7333,381],{"className":7334},[415,419,450],[394,7336,457],{"className":7337},[516,450],[394,7339,520],{"className":7340},[415,450],[394,7342],{"className":7343},[465,524],[394,7345],{"className":7346,"style":528},[465],[394,7348,7350,7353,7356,7365,7368,7371],{"className":7349},[406],[394,7351],{"className":7352,"style":535},[410],[394,7354,540],{"className":7355},[539],[394,7357,7359],{"className":7358},[415],[394,7360,7362],{"className":7361},[415],[394,7363,551],{"className":7364},[415,550],[394,7366],{"className":7367,"style":555},[465],[394,7369,381],{"className":7370},[415,419],[394,7372,563],{"className":7373},[562]," for any ",[394,7376,7378],{"className":7377},[397],[394,7379,7381,7436],{"className":7380,"ariaHidden":402},[401],[394,7382,7384,7387,7390,7393,7426,7430,7433],{"className":7383},[406],[394,7385],{"className":7386,"style":761},[410],[394,7388,384],{"className":7389},[415,419],[394,7391],{"className":7392,"style":466},[465],[394,7394,7396],{"className":7395},[470],[394,7397,7400],{"className":7398},[415,7399],"vbox",[394,7401,7404],{"className":7402},[7403],"thinbox",[394,7405,7408,7411,7422],{"className":7406},[7407],"rlap",[394,7409],{"className":7410,"style":761},[410],[394,7412,7415],{"className":7413},[7414],"inner",[394,7416,7418],{"className":7417},[415],[394,7419,7421],{"className":7420},[470],"",[394,7423],{"className":7424},[7425],"fix",[394,7427],{"className":7428},[465,7429],"nobreak",[394,7431,471],{"className":7432},[470],[394,7434],{"className":7435,"style":466},[465],[394,7437,7439,7442],{"className":7438},[406],[394,7440],{"className":7441,"style":639},[410],[394,7443,780],{"className":7444},[415],", computed by the\nmodular exponentiation routine.",[381,7447,7448,7449,7531,7532,7569,7570,392,7573,7690],{},"The plan: precompute the factorials ",[394,7450,7452],{"className":7451},[397],[394,7453,7455,7489,7522],{"className":7454,"ariaHidden":402},[401],[394,7456,7458,7461,7469,7473,7476,7480,7483,7486],{"className":7457},[406],[394,7459],{"className":7460,"style":535},[410],[394,7462,7465],{"className":7463},[415,7464],"text",[394,7466,7468],{"className":7467},[415],"fact",[394,7470,7472],{"className":7471},[539],"[",[394,7474,5212],{"className":7475},[415,419],[394,7477,7479],{"className":7478},[562],"]",[394,7481],{"className":7482,"style":466},[465],[394,7484,674],{"className":7485},[470],[394,7487],{"className":7488,"style":466},[465],[394,7490,7492,7495,7498,7501,7504,7507,7516,7519],{"className":7491},[406],[394,7493],{"className":7494,"style":660},[410],[394,7496,5212],{"className":7497},[415,419],[394,7499,667],{"className":7500},[562],[394,7502],{"className":7503,"style":7042},[465],[394,7505],{"className":7506,"style":626},[465],[394,7508,7510],{"className":7509},[516],[394,7511,7513],{"className":7512},[415],[394,7514,551],{"className":7515},[415,550],[394,7517],{"className":7518,"style":7042},[465],[394,7520],{"className":7521,"style":626},[465],[394,7523,7525,7528],{"className":7524},[406],[394,7526],{"className":7527,"style":4597},[410],[394,7529,381],{"className":7530},[415,419]," for all ",[394,7533,7535],{"className":7534},[397],[394,7536,7538,7558],{"className":7537,"ariaHidden":402},[401],[394,7539,7541,7545,7548,7551,7555],{"className":7540},[406],[394,7542],{"className":7543,"style":7544},[410],"height:0.7955em;vertical-align:-0.136em;",[394,7546,5212],{"className":7547},[415,419],[394,7549],{"className":7550,"style":466},[465],[394,7552,7554],{"className":7553},[470],"≤",[394,7556],{"className":7557,"style":466},[465],[394,7559,7561,7564],{"className":7560},[406],[394,7562],{"className":7563,"style":871},[410],[394,7565,7568],{"className":7566,"style":7567},[415,419],"margin-right:0.109em;","N",",\nand the ",[389,7571,7572],{},"inverse factorials",[394,7574,7576],{"className":7575},[397],[394,7577,7579,7610,7681],{"className":7578,"ariaHidden":402},[401],[394,7580,7582,7585,7592,7595,7598,7601,7604,7607],{"className":7581},[406],[394,7583],{"className":7584,"style":535},[410],[394,7586,7588],{"className":7587},[415,7464],[394,7589,7591],{"className":7590},[415],"invfact",[394,7593,7472],{"className":7594},[539],[394,7596,5212],{"className":7597},[415,419],[394,7599,7479],{"className":7600},[562],[394,7602],{"className":7603,"style":466},[465],[394,7605,674],{"className":7606},[470],[394,7608],{"className":7609,"style":466},[465],[394,7611,7613,7616,7619,7622,7625,7660,7663,7666,7675,7678],{"className":7612},[406],[394,7614],{"className":7615,"style":3596},[410],[394,7617,540],{"className":7618},[539],[394,7620,5212],{"className":7621},[415,419],[394,7623,667],{"className":7624},[562],[394,7626,7628,7631],{"className":7627},[562],[394,7629,563],{"className":7630},[562],[394,7632,7634],{"className":7633},[423],[394,7635,7637],{"className":7636},[427],[394,7638,7640],{"className":7639},[431],[394,7641,7643],{"className":7642,"style":411},[435],[394,7644,7645,7648],{"style":438},[394,7646],{"className":7647,"style":443},[442],[394,7649,7651],{"className":7650},[447,448,449,450],[394,7652,7654,7657],{"className":7653},[415,450],[394,7655,457],{"className":7656},[415,450],[394,7658,461],{"className":7659},[415,450],[394,7661],{"className":7662,"style":7042},[465],[394,7664],{"className":7665,"style":626},[465],[394,7667,7669],{"className":7668},[516],[394,7670,7672],{"className":7671},[415],[394,7673,551],{"className":7674},[415,550],[394,7676],{"className":7677,"style":7042},[465],[394,7679],{"className":7680,"style":626},[465],[394,7682,7684,7687],{"className":7683},[406],[394,7685],{"className":7686,"style":4597},[410],[394,7688,381],{"className":7689},[415,419],". With both\ntables in hand, every binomial coefficient is a single product.",[7692,7693,7697],"pre",{"className":7694,"code":7695,"language":7696,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Precompute-Factorials}(N, p)$ — $O(N)$ tables for $O(1)$ queries\n$\\text{fact}[0] \\gets 1$\nfor $i \\gets 1$ to $N$ do\n  $\\text{fact}[i] \\gets \\text{fact}[i-1] \\cdot i \\bmod p$\n$\\text{invfact}[N] \\gets \\textsc{Mod-Pow}(\\text{fact}[N],\\, p-2,\\, p)$ \u002F\u002F one Fermat inverse\nfor $i \\gets N$ downto $1$ do\n  $\\text{invfact}[i-1] \\gets \\text{invfact}[i] \\cdot i \\bmod p$ \u002F\u002F peel a factor\n","algorithm",[7698,7699,7700,7706,7711,7716,7721,7726,7731],"code",{"__ignoreMap":376},[394,7701,7703],{"class":7702,"line":6},"line",[394,7704,7705],{},"caption: $\\textsc{Precompute-Factorials}(N, p)$ — $O(N)$ tables for $O(1)$ queries\n",[394,7707,7708],{"class":7702,"line":18},[394,7709,7710],{},"$\\text{fact}[0] \\gets 1$\n",[394,7712,7713],{"class":7702,"line":24},[394,7714,7715],{},"for $i \\gets 1$ to $N$ do\n",[394,7717,7718],{"class":7702,"line":73},[394,7719,7720],{},"  $\\text{fact}[i] \\gets \\text{fact}[i-1] \\cdot i \\bmod p$\n",[394,7722,7723],{"class":7702,"line":102},[394,7724,7725],{},"$\\text{invfact}[N] \\gets \\textsc{Mod-Pow}(\\text{fact}[N],\\, p-2,\\, p)$ \u002F\u002F one Fermat inverse\n",[394,7727,7728],{"class":7702,"line":108},[394,7729,7730],{},"for $i \\gets N$ downto $1$ do\n",[394,7732,7733],{"class":7702,"line":116},[394,7734,7735],{},"  $\\text{invfact}[i-1] \\gets \\text{invfact}[i] \\cdot i \\bmod p$ \u002F\u002F peel a factor\n",[381,7737,7738,7739,7763,7764,7805,7806,7809,7810,7837,7838,7983],{},"The downward loop is the trick that keeps the precompute at ",[394,7740,7742],{"className":7741},[397],[394,7743,7745],{"className":7744,"ariaHidden":402},[401],[394,7746,7748,7751,7754,7757,7760],{"className":7747},[406],[394,7749],{"className":7750,"style":535},[410],[394,7752,3600],{"className":7753,"style":821},[415,419],[394,7755,540],{"className":7756},[539],[394,7758,7568],{"className":7759,"style":7567},[415,419],[394,7761,563],{"className":7762},[562]," rather than\n",[394,7765,7767],{"className":7766},[397],[394,7768,7770],{"className":7769,"ariaHidden":402},[401],[394,7771,7773,7776,7779,7782,7785,7788,7796,7799,7802],{"className":7772},[406],[394,7774],{"className":7775,"style":535},[410],[394,7777,3600],{"className":7778,"style":821},[415,419],[394,7780,540],{"className":7781},[539],[394,7783,7568],{"className":7784,"style":7567},[415,419],[394,7786],{"className":7787,"style":734},[465],[394,7789,7791],{"className":7790},[4314],[394,7792,7795],{"className":7793,"style":7794},[415,550],"margin-right:0.0139em;","log",[394,7797],{"className":7798,"style":734},[465],[394,7800,381],{"className":7801},[415,419],[394,7803,563],{"className":7804},[562],": only ",[389,7807,7808],{},"one"," modular exponentiation is needed, for ",[394,7811,7813],{"className":7812},[397],[394,7814,7816],{"className":7815,"ariaHidden":402},[401],[394,7817,7819,7822,7828,7831,7834],{"className":7818},[406],[394,7820],{"className":7821,"style":535},[410],[394,7823,7825],{"className":7824},[415,7464],[394,7826,7591],{"className":7827},[415],[394,7829,7472],{"className":7830},[539],[394,7832,7568],{"className":7833,"style":7567},[415,419],[394,7835,7479],{"className":7836},[562],";\neach smaller inverse factorial follows from ",[394,7839,7841],{"className":7840},[397],[394,7842,7844,7865,7921,7974],{"className":7843,"ariaHidden":402},[401],[394,7845,7847,7850,7853,7856,7859,7862],{"className":7846},[406],[394,7848],{"className":7849,"style":535},[410],[394,7851,540],{"className":7852},[539],[394,7854,5212],{"className":7855},[415,419],[394,7857],{"className":7858,"style":626},[465],[394,7860,457],{"className":7861},[516],[394,7863],{"className":7864,"style":626},[465],[394,7866,7868,7871,7874,7877,7912,7915,7918],{"className":7867},[406],[394,7869],{"className":7870,"style":3596},[410],[394,7872,461],{"className":7873},[415],[394,7875,563],{"className":7876},[562],[394,7878,7880,7883],{"className":7879},[562],[394,7881,667],{"className":7882},[562],[394,7884,7886],{"className":7885},[423],[394,7887,7889],{"className":7888},[427],[394,7890,7892],{"className":7891},[431],[394,7893,7895],{"className":7894,"style":411},[435],[394,7896,7897,7900],{"style":438},[394,7898],{"className":7899,"style":443},[442],[394,7901,7903],{"className":7902},[447,448,449,450],[394,7904,7906,7909],{"className":7905},[415,450],[394,7907,457],{"className":7908},[415,450],[394,7910,461],{"className":7911},[415,450],[394,7913],{"className":7914,"style":466},[465],[394,7916,674],{"className":7917},[470],[394,7919],{"className":7920,"style":466},[465],[394,7922,7924,7927,7930,7965,7968,7971],{"className":7923},[406],[394,7925],{"className":7926,"style":411},[410],[394,7928,5212],{"className":7929},[415,419],[394,7931,7933,7936],{"className":7932},[562],[394,7934,667],{"className":7935},[562],[394,7937,7939],{"className":7938},[423],[394,7940,7942],{"className":7941},[427],[394,7943,7945],{"className":7944},[431],[394,7946,7948],{"className":7947,"style":411},[435],[394,7949,7950,7953],{"style":438},[394,7951],{"className":7952,"style":443},[442],[394,7954,7956],{"className":7955},[447,448,449,450],[394,7957,7959,7962],{"className":7958},[415,450],[394,7960,457],{"className":7961},[415,450],[394,7963,461],{"className":7964},[415,450],[394,7966],{"className":7967,"style":626},[465],[394,7969,694],{"className":7970},[516],[394,7972],{"className":7973,"style":626},[465],[394,7975,7977,7980],{"className":7976},[406],[394,7978],{"className":7979,"style":5482},[410],[394,7981,5212],{"className":7982},[415,419],". Then each\nquery is constant time:",[394,7985,7987],{"className":7986},[647],[394,7988,7990],{"className":7989},[397],[394,7991,7993,8068,8098,8128,8155,8173,8221,8240,8259],{"className":7992,"ariaHidden":402},[401],[394,7994,7996,7999,8059,8062,8065],{"className":7995},[406],[394,7997],{"className":7998,"style":1218},[410],[394,8000,8002,8008,8053],{"className":8001},[415],[394,8003,8005],{"className":8004,"style":1226},[539,1225],[394,8006,540],{"className":8007},[1230,449],[394,8009,8011],{"className":8010},[909],[394,8012,8014,8045],{"className":8013},[427,913],[394,8015,8017,8042],{"className":8016},[431],[394,8018,8020,8031],{"className":8019,"style":1243},[435],[394,8021,8022,8025],{"style":923},[394,8023],{"className":8024,"style":927},[442],[394,8026,8028],{"className":8027},[415],[394,8029,2068],{"className":8030,"style":2067},[415,419],[394,8032,8033,8036],{"style":966},[394,8034],{"className":8035,"style":927},[442],[394,8037,8039],{"className":8038},[415],[394,8040,605],{"className":8041},[415,419],[394,8043,983],{"className":8044},[982],[394,8046,8048],{"className":8047},[431],[394,8049,8051],{"className":8050,"style":1275},[435],[394,8052],{},[394,8054,8056],{"className":8055,"style":1226},[562,1225],[394,8057,563],{"className":8058},[1230,449],[394,8060],{"className":8061,"style":466},[465],[394,8063,471],{"className":8064},[470],[394,8066],{"className":8067,"style":466},[465],[394,8069,8071,8074,8080,8083,8086,8089,8092,8095],{"className":8070},[406],[394,8072],{"className":8073,"style":535},[410],[394,8075,8077],{"className":8076},[415,7464],[394,8078,7468],{"className":8079},[415],[394,8081,7472],{"className":8082},[539],[394,8084,605],{"className":8085},[415,419],[394,8087,7479],{"className":8088},[562],[394,8090],{"className":8091,"style":626},[465],[394,8093,694],{"className":8094},[516],[394,8096],{"className":8097,"style":626},[465],[394,8099,8101,8104,8110,8113,8116,8119,8122,8125],{"className":8100},[406],[394,8102],{"className":8103,"style":535},[410],[394,8105,8107],{"className":8106},[415,7464],[394,8108,7591],{"className":8109},[415],[394,8111,7472],{"className":8112},[539],[394,8114,2068],{"className":8115,"style":2067},[415,419],[394,8117,7479],{"className":8118},[562],[394,8120],{"className":8121,"style":626},[465],[394,8123,694],{"className":8124},[516],[394,8126],{"className":8127,"style":626},[465],[394,8129,8131,8134,8140,8143,8146,8149,8152],{"className":8130},[406],[394,8132],{"className":8133,"style":535},[410],[394,8135,8137],{"className":8136},[415,7464],[394,8138,7591],{"className":8139},[415],[394,8141,7472],{"className":8142},[539],[394,8144,605],{"className":8145},[415,419],[394,8147],{"className":8148,"style":626},[465],[394,8150,457],{"className":8151},[516],[394,8153],{"className":8154,"style":626},[465],[394,8156,8158,8161,8164,8167,8170],{"className":8157},[406],[394,8159],{"className":8160,"style":535},[410],[394,8162,2068],{"className":8163,"style":2067},[415,419],[394,8165,7479],{"className":8166},[562],[394,8168],{"className":8169},[465,524],[394,8171],{"className":8172,"style":5694},[465],[394,8174,8176,8179,8182,8191,8194,8197,8200,8203,8206,8209,8212,8215,8218],{"className":8175},[406],[394,8177],{"className":8178,"style":535},[410],[394,8180,540],{"className":8181},[539],[394,8183,8185],{"className":8184},[415],[394,8186,8188],{"className":8187},[415],[394,8189,551],{"className":8190},[415,550],[394,8192],{"className":8193,"style":555},[465],[394,8195,381],{"className":8196},[415,419],[394,8198,563],{"className":8199},[562],[394,8201,769],{"className":8202},[768],[394,8204],{"className":8205,"style":773},[465],[394,8207],{"className":8208,"style":734},[465],[394,8210,780],{"className":8211},[415],[394,8213],{"className":8214,"style":466},[465],[394,8216,7554],{"className":8217},[470],[394,8219],{"className":8220,"style":466},[465],[394,8222,8224,8228,8231,8234,8237],{"className":8223},[406],[394,8225],{"className":8226,"style":8227},[410],"height:0.8304em;vertical-align:-0.136em;",[394,8229,2068],{"className":8230,"style":2067},[415,419],[394,8232],{"className":8233,"style":466},[465],[394,8235,7554],{"className":8236},[470],[394,8238],{"className":8239,"style":466},[465],[394,8241,8243,8247,8250,8253,8256],{"className":8242},[406],[394,8244],{"className":8245,"style":8246},[410],"height:0.7719em;vertical-align:-0.136em;",[394,8248,605],{"className":8249},[415,419],[394,8251],{"className":8252,"style":466},[465],[394,8254,7554],{"className":8255},[470],[394,8257],{"className":8258,"style":466},[465],[394,8260,8262,8265,8268],{"className":8261},[406],[394,8263],{"className":8264,"style":871},[410],[394,8266,7568],{"className":8267,"style":7567},[415,419],[394,8269,1099],{"className":8270},[415],[381,8272,8273,8274,8298,8299,8323,8324,6249,8327,8330,8331,1099],{},"This ",[394,8275,8277],{"className":8276},[397],[394,8278,8280],{"className":8279,"ariaHidden":402},[401],[394,8281,8283,8286,8289,8292,8295],{"className":8282},[406],[394,8284],{"className":8285,"style":535},[410],[394,8287,3600],{"className":8288,"style":821},[415,419],[394,8290,540],{"className":8291},[539],[394,8293,7568],{"className":8294,"style":7567},[415,419],[394,8296,563],{"className":8297},[562],"-precompute, ",[394,8300,8302],{"className":8301},[397],[394,8303,8305],{"className":8304,"ariaHidden":402},[401],[394,8306,8308,8311,8314,8317,8320],{"className":8307},[406],[394,8309],{"className":8310,"style":535},[410],[394,8312,3600],{"className":8313,"style":821},[415,419],[394,8315,540],{"className":8316},[539],[394,8318,461],{"className":8319},[415],[394,8321,563],{"className":8322},[562],"-query scheme is exactly what ",[1107,8325,8326],{},"Number of Music\nPlaylists",[1107,8328,8329],{},"Count Anagrams"," need, since both reduce to products and ratios of\nfactorials modulo ",[394,8332,8334],{"className":8333},[397],[394,8335,8337,8384],{"className":8336,"ariaHidden":402},[401],[394,8338,8340,8343,8346,8375,8378,8381],{"className":8339},[406],[394,8341],{"className":8342,"style":7101},[410],[394,8344,461],{"className":8345},[415],[394,8347,8349,8352],{"className":8348},[415],[394,8350,780],{"className":8351},[415],[394,8353,8355],{"className":8354},[423],[394,8356,8358],{"className":8357},[427],[394,8359,8361],{"className":8360},[431],[394,8362,8364],{"className":8363,"style":411},[435],[394,8365,8366,8369],{"style":438},[394,8367],{"className":8368,"style":443},[442],[394,8370,8372],{"className":8371},[447,448,449,450],[394,8373,7134],{"className":8374},[415,450],[394,8376],{"className":8377,"style":626},[465],[394,8379,1080],{"className":8380},[516],[394,8382],{"className":8383,"style":626},[465],[394,8385,8387,8390],{"className":8386},[406],[394,8388],{"className":8389,"style":639},[410],[394,8391,6353],{"className":8392},[415],[2022,8394,8396],{"type":8395},"theorem",[381,8397,8398,8401,8402,6249,8417,8432,8433,8448,8449,8464,8465,8480,8481,6249,8588,2087,8694,9013,9014,1099,9029],{},[389,8399,8400],{},"Theorem (Lucas')."," When ",[394,8403,8405],{"className":8404},[397],[394,8406,8408],{"className":8407,"ariaHidden":402},[401],[394,8409,8411,8414],{"className":8410},[406],[394,8412],{"className":8413,"style":601},[410],[394,8415,605],{"className":8416},[415,419],[394,8418,8420],{"className":8419},[397],[394,8421,8423],{"className":8422,"ariaHidden":402},[401],[394,8424,8426,8429],{"className":8425},[406],[394,8427],{"className":8428,"style":660},[410],[394,8430,2068],{"className":8431,"style":2067},[415,419]," can exceed ",[394,8434,8436],{"className":8435},[397],[394,8437,8439],{"className":8438,"ariaHidden":402},[401],[394,8440,8442,8445],{"className":8441},[406],[394,8443],{"className":8444,"style":4597},[410],[394,8446,381],{"className":8447},[415,419]," itself, the factorial tables\nbreak (they contain factors of ",[394,8450,8452],{"className":8451},[397],[394,8453,8455],{"className":8454,"ariaHidden":402},[401],[394,8456,8458,8461],{"className":8457},[406],[394,8459],{"className":8460,"style":4597},[410],[394,8462,381],{"className":8463},[415,419],", hence zeros). Lucas reduces the problem to\ndigits in base ",[394,8466,8468],{"className":8467},[397],[394,8469,8471],{"className":8470,"ariaHidden":402},[401],[394,8472,8474,8477],{"className":8473},[406],[394,8475],{"className":8476,"style":4597},[410],[394,8478,381],{"className":8479},[415,419],": writing ",[394,8482,8484],{"className":8483},[397],[394,8485,8487,8505],{"className":8486,"ariaHidden":402},[401],[394,8488,8490,8493,8496,8499,8502],{"className":8489},[406],[394,8491],{"className":8492,"style":601},[410],[394,8494,605],{"className":8495},[415,419],[394,8497],{"className":8498,"style":466},[465],[394,8500,674],{"className":8501},[470],[394,8503],{"className":8504,"style":466},[465],[394,8506,8508,8512,8515,8518,8558],{"className":8507},[406],[394,8509],{"className":8510,"style":8511},[410],"height:1.0747em;vertical-align:-0.25em;",[394,8513,4361],{"className":8514,"style":4752},[4314,4359,4751],[394,8516],{"className":8517,"style":734},[465],[394,8519,8521,8524],{"className":8520},[415],[394,8522,605],{"className":8523},[415,419],[394,8525,8527],{"className":8526},[423],[394,8528,8530,8550],{"className":8529},[427,913],[394,8531,8533,8547],{"className":8532},[431],[394,8534,8536],{"className":8535,"style":5200},[435],[394,8537,8538,8541],{"style":5016},[394,8539],{"className":8540,"style":443},[442],[394,8542,8544],{"className":8543},[447,448,449,450],[394,8545,5212],{"className":8546},[415,419,450],[394,8548,983],{"className":8549},[982],[394,8551,8553],{"className":8552},[431],[394,8554,8556],{"className":8555,"style":5035},[435],[394,8557],{},[394,8559,8561,8564],{"className":8560},[415],[394,8562,381],{"className":8563},[415,419],[394,8565,8567],{"className":8566},[423],[394,8568,8570],{"className":8569},[427],[394,8571,8573],{"className":8572},[431],[394,8574,8577],{"className":8575,"style":8576},[435],"height:0.8247em;",[394,8578,8579,8582],{"style":438},[394,8580],{"className":8581,"style":443},[442],[394,8583,8585],{"className":8584},[447,448,449,450],[394,8586,5212],{"className":8587},[415,419,450],[394,8589,8591],{"className":8590},[397],[394,8592,8594,8612],{"className":8593,"ariaHidden":402},[401],[394,8595,8597,8600,8603,8606,8609],{"className":8596},[406],[394,8598],{"className":8599,"style":660},[410],[394,8601,2068],{"className":8602,"style":2067},[415,419],[394,8604],{"className":8605,"style":466},[465],[394,8607,674],{"className":8608},[470],[394,8610],{"className":8611,"style":466},[465],[394,8613,8615,8618,8621,8624,8665],{"className":8614},[406],[394,8616],{"className":8617,"style":8511},[410],[394,8619,4361],{"className":8620,"style":4752},[4314,4359,4751],[394,8622],{"className":8623,"style":734},[465],[394,8625,8627,8630],{"className":8626},[415],[394,8628,2068],{"className":8629,"style":2067},[415,419],[394,8631,8633],{"className":8632},[423],[394,8634,8636,8657],{"className":8635},[427,913],[394,8637,8639,8654],{"className":8638},[431],[394,8640,8642],{"className":8641,"style":5200},[435],[394,8643,8645,8648],{"style":8644},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[394,8646],{"className":8647,"style":443},[442],[394,8649,8651],{"className":8650},[447,448,449,450],[394,8652,5212],{"className":8653},[415,419,450],[394,8655,983],{"className":8656},[982],[394,8658,8660],{"className":8659},[431],[394,8661,8663],{"className":8662,"style":5035},[435],[394,8664],{},[394,8666,8668,8671],{"className":8667},[415],[394,8669,381],{"className":8670},[415,419],[394,8672,8674],{"className":8673},[423],[394,8675,8677],{"className":8676},[427],[394,8678,8680],{"className":8679},[431],[394,8681,8683],{"className":8682,"style":8576},[435],[394,8684,8685,8688],{"style":438},[394,8686],{"className":8687,"style":443},[442],[394,8689,8691],{"className":8690},[447,448,449,450],[394,8692,5212],{"className":8693},[415,419,450],[394,8695,8697],{"className":8696},[397],[394,8698,8700,8781,8986],{"className":8699,"ariaHidden":402},[401],[394,8701,8703,8706,8772,8775,8778],{"className":8702},[406],[394,8704],{"className":8705,"style":1497},[410],[394,8707,8709,8715,8766],{"className":8708},[415],[394,8710,8712],{"className":8711,"style":1226},[539,1225],[394,8713,540],{"className":8714},[1230,1507],[394,8716,8718],{"className":8717},[909],[394,8719,8721,8758],{"className":8720},[427,913],[394,8722,8724,8755],{"className":8723},[431],[394,8725,8727,8741],{"className":8726,"style":1520},[435],[394,8728,8729,8732],{"style":1523},[394,8730],{"className":8731,"style":443},[442],[394,8733,8735],{"className":8734},[447,448,449,450],[394,8736,8738],{"className":8737},[415,450],[394,8739,2068],{"className":8740,"style":2067},[415,419,450],[394,8742,8743,8746],{"style":1538},[394,8744],{"className":8745,"style":443},[442],[394,8747,8749],{"className":8748},[447,448,449,450],[394,8750,8752],{"className":8751},[415,450],[394,8753,605],{"className":8754},[415,419,450],[394,8756,983],{"className":8757},[982],[394,8759,8761],{"className":8760},[431],[394,8762,8764],{"className":8763,"style":1560},[435],[394,8765],{},[394,8767,8769],{"className":8768,"style":1226},[562,1225],[394,8770,563],{"className":8771},[1230,1507],[394,8773],{"className":8774,"style":466},[465],[394,8776,471],{"className":8777},[470],[394,8779],{"className":8780,"style":466},[465],[394,8782,8784,8788,8830,8833,8980,8983],{"className":8783},[406],[394,8785],{"className":8786,"style":8787},[410],"height:1.2951em;vertical-align:-0.4451em;",[394,8789,8791,8795],{"className":8790},[4314],[394,8792,8794],{"className":8793,"style":4752},[4314,4359,4751],"∏",[394,8796,8798],{"className":8797},[423],[394,8799,8801,8822],{"className":8800},[427,913],[394,8802,8804,8819],{"className":8803},[431],[394,8805,8808],{"className":8806,"style":8807},[435],"height:0.162em;",[394,8809,8810,8813],{"style":4768},[394,8811],{"className":8812,"style":443},[442],[394,8814,8816],{"className":8815},[447,448,449,450],[394,8817,5212],{"className":8818},[415,419,450],[394,8820,983],{"className":8821},[982],[394,8823,8825],{"className":8824},[431],[394,8826,8828],{"className":8827,"style":4787},[435],[394,8829],{},[394,8831],{"className":8832,"style":734},[465],[394,8834,8836,8842,8974],{"className":8835},[415],[394,8837,8839],{"className":8838,"style":1226},[539,1225],[394,8840,540],{"className":8841},[1230,1507],[394,8843,8845],{"className":8844},[909],[394,8846,8848,8965],{"className":8847},[427,913],[394,8849,8851,8962],{"className":8850},[431],[394,8852,8854,8910],{"className":8853,"style":1520},[435],[394,8855,8856,8859],{"style":1523},[394,8857],{"className":8858,"style":443},[442],[394,8860,8862],{"className":8861},[447,448,449,450],[394,8863,8865],{"className":8864},[415,450],[394,8866,8868,8871],{"className":8867},[415,450],[394,8869,2068],{"className":8870,"style":2067},[415,419,450],[394,8872,8874],{"className":8873},[423],[394,8875,8877,8901],{"className":8876},[427,913],[394,8878,8880,8898],{"className":8879},[431],[394,8881,8884],{"className":8882,"style":8883},[435],"height:0.3281em;",[394,8885,8887,8891],{"style":8886},"top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;",[394,8888],{"className":8889,"style":8890},[442],"height:2.5em;",[394,8892,8895],{"className":8893},[447,8894,1507,450],"reset-size3",[394,8896,5212],{"className":8897},[415,419,450],[394,8899,983],{"className":8900},[982],[394,8902,8904],{"className":8903},[431],[394,8905,8908],{"className":8906,"style":8907},[435],"height:0.143em;",[394,8909],{},[394,8911,8912,8915],{"style":1538},[394,8913],{"className":8914,"style":443},[442],[394,8916,8918],{"className":8917},[447,448,449,450],[394,8919,8921],{"className":8920},[415,450],[394,8922,8924,8927],{"className":8923},[415,450],[394,8925,605],{"className":8926},[415,419,450],[394,8928,8930],{"className":8929},[423],[394,8931,8933,8954],{"className":8932},[427,913],[394,8934,8936,8951],{"className":8935},[431],[394,8937,8939],{"className":8938,"style":8883},[435],[394,8940,8942,8945],{"style":8941},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[394,8943],{"className":8944,"style":8890},[442],[394,8946,8948],{"className":8947},[447,8894,1507,450],[394,8949,5212],{"className":8950},[415,419,450],[394,8952,983],{"className":8953},[982],[394,8955,8957],{"className":8956},[431],[394,8958,8960],{"className":8959,"style":8907},[435],[394,8961],{},[394,8963,983],{"className":8964},[982],[394,8966,8968],{"className":8967},[431],[394,8969,8972],{"className":8970,"style":8971},[435],"height:0.4451em;",[394,8973],{},[394,8975,8977],{"className":8976,"style":1226},[562,1225],[394,8978,563],{"className":8979},[1230,1507],[394,8981],{"className":8982},[465,524],[394,8984],{"className":8985,"style":528},[465],[394,8987,8989,8992,8995,9004,9007,9010],{"className":8988},[406],[394,8990],{"className":8991,"style":535},[410],[394,8993,540],{"className":8994},[539],[394,8996,8998],{"className":8997},[415],[394,8999,9001],{"className":9000},[415],[394,9002,551],{"className":9003},[415,550],[394,9005],{"className":9006,"style":555},[465],[394,9008,381],{"className":9009},[415,419],[394,9011,563],{"className":9012},[562],", a product of small binomial\ncoefficients each computable from a table of size ",[394,9015,9017],{"className":9016},[397],[394,9018,9020],{"className":9019,"ariaHidden":402},[401],[394,9021,9023,9026],{"className":9022},[406],[394,9024],{"className":9025,"style":4597},[410],[394,9027,381],{"className":9028},[415,419],[1609,9030,9031],{},[384,9032,520],{"href":9033,"ariaDescribedBy":9034,"dataFootnoteRef":376,"id":9035},"#user-content-fn-lucas",[1615],"user-content-fnref-lucas",[578,9037,9039],{"id":9038},"inclusionexclusion","Inclusion–exclusion",[381,9041,9042,9043,9046],{},"To count a ",[389,9044,9045],{},"union"," of overlapping sets we cannot simply add their sizes, since elements\nin several sets get counted several times. Inclusion–exclusion corrects the overcount\nwith alternating signs:",[394,9048,9050],{"className":9049},[647],[394,9051,9053],{"className":9052},[397],[394,9054,9056,9276,9392,9511,9577,9699,9755,9819,9843],{"className":9055,"ariaHidden":402},[401],[394,9057,9059,9063,9110,9113,9183,9186,9227,9230,9267,9270,9273],{"className":9058},[406],[394,9060],{"className":9061,"style":9062},[410],"height:2.9291em;vertical-align:-1.2777em;",[394,9064,9066],{"className":9065},[539],[394,9067,9070],{"className":9068},[1230,9069],"mult",[394,9071,9073,9101],{"className":9072},[427,913],[394,9074,9076,9098],{"className":9075},[431],[394,9077,9080],{"className":9078,"style":9079},[435],"height:1.15em;",[394,9081,9083,9087],{"style":9082},"top:-3.15em;",[394,9084],{"className":9085,"style":9086},[442],"height:3.8em;",[394,9088,9090],{"style":9089},"width:0.333em;height:1.8em;",[2823,9091,9095],{"xmlns":2825,"width":9092,"height":9093,"viewBox":9094},"0.333em","1.8em","0 0 333 1800",[2835,9096],{"d":9097},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[394,9099,983],{"className":9100},[982],[394,9102,9104],{"className":9103},[431],[394,9105,9108],{"className":9106,"style":9107},[435],"height:0.65em;",[394,9109],{},[394,9111],{"className":9112,"style":734},[465],[394,9114,9116],{"className":9115},[4314,4315],[394,9117,9119,9174],{"className":9118},[427,913],[394,9120,9122,9171],{"className":9121},[431],[394,9123,9125,9146,9157],{"className":9124,"style":4325},[435],[394,9126,9128,9131],{"style":9127},"top:-1.8723em;margin-left:0em;",[394,9129],{"className":9130,"style":4332},[442],[394,9132,9134],{"className":9133},[447,448,449,450],[394,9135,9137,9140,9143],{"className":9136},[415,450],[394,9138,5212],{"className":9139},[415,419,450],[394,9141,674],{"className":9142},[470,450],[394,9144,461],{"className":9145},[415,450],[394,9147,9148,9151],{"style":4350},[394,9149],{"className":9150,"style":4332},[442],[394,9152,9153],{},[394,9154,9156],{"className":9155},[4314,4359,4360],"⋃",[394,9158,9159,9162],{"style":4364},[394,9160],{"className":9161,"style":4332},[442],[394,9163,9165],{"className":9164},[447,448,449,450],[394,9166,9168],{"className":9167},[415,450],[394,9169,605],{"className":9170},[415,419,450],[394,9172,983],{"className":9173},[982],[394,9175,9177],{"className":9176},[431],[394,9178,9181],{"className":9179,"style":9180},[435],"height:1.2777em;",[394,9182],{},[394,9184],{"className":9185,"style":734},[465],[394,9187,9189,9193],{"className":9188},[415],[394,9190,9192],{"className":9191},[415,419],"A",[394,9194,9196],{"className":9195},[423],[394,9197,9199,9219],{"className":9198},[427,913],[394,9200,9202,9216],{"className":9201},[431],[394,9203,9205],{"className":9204,"style":5200},[435],[394,9206,9207,9210],{"style":5016},[394,9208],{"className":9209,"style":443},[442],[394,9211,9213],{"className":9212},[447,448,449,450],[394,9214,5212],{"className":9215},[415,419,450],[394,9217,983],{"className":9218},[982],[394,9220,9222],{"className":9221},[431],[394,9223,9225],{"className":9224,"style":5035},[435],[394,9226],{},[394,9228],{"className":9229,"style":734},[465],[394,9231,9233],{"className":9232},[562],[394,9234,9236],{"className":9235},[1230,9069],[394,9237,9239,9259],{"className":9238},[427,913],[394,9240,9242,9256],{"className":9241},[431],[394,9243,9245],{"className":9244,"style":9079},[435],[394,9246,9247,9250],{"style":9082},[394,9248],{"className":9249,"style":9086},[442],[394,9251,9252],{"style":9089},[2823,9253,9254],{"xmlns":2825,"width":9092,"height":9093,"viewBox":9094},[2835,9255],{"d":9097},[394,9257,983],{"className":9258},[982],[394,9260,9262],{"className":9261},[431],[394,9263,9265],{"className":9264,"style":9107},[435],[394,9266],{},[394,9268],{"className":9269,"style":466},[465],[394,9271,674],{"className":9272},[470],[394,9274],{"className":9275,"style":466},[465],[394,9277,9279,9283,9328,9331,9334,9374,9377,9380,9383,9386,9389],{"className":9278},[406],[394,9280],{"className":9281,"style":9282},[410],"height:2.3277em;vertical-align:-1.2777em;",[394,9284,9286],{"className":9285},[4314,4315],[394,9287,9289,9320],{"className":9288},[427,913],[394,9290,9292,9317],{"className":9291},[431],[394,9293,9296,9307],{"className":9294,"style":9295},[435],"height:1.05em;",[394,9297,9298,9301],{"style":9127},[394,9299],{"className":9300,"style":4332},[442],[394,9302,9304],{"className":9303},[447,448,449,450],[394,9305,5212],{"className":9306},[415,419,450],[394,9308,9309,9312],{"style":4350},[394,9310],{"className":9311,"style":4332},[442],[394,9313,9314],{},[394,9315,4361],{"className":9316},[4314,4359,4360],[394,9318,983],{"className":9319},[982],[394,9321,9323],{"className":9322},[431],[394,9324,9326],{"className":9325,"style":9180},[435],[394,9327],{},[394,9329],{"className":9330,"style":734},[465],[394,9332,5640],{"className":9333},[415],[394,9335,9337,9340],{"className":9336},[415],[394,9338,9192],{"className":9339},[415,419],[394,9341,9343],{"className":9342},[423],[394,9344,9346,9366],{"className":9345},[427,913],[394,9347,9349,9363],{"className":9348},[431],[394,9350,9352],{"className":9351,"style":5200},[435],[394,9353,9354,9357],{"style":5016},[394,9355],{"className":9356,"style":443},[442],[394,9358,9360],{"className":9359},[447,448,449,450],[394,9361,5212],{"className":9362},[415,419,450],[394,9364,983],{"className":9365},[982],[394,9367,9369],{"className":9368},[431],[394,9370,9372],{"className":9371,"style":5035},[435],[394,9373],{},[394,9375,5640],{"className":9376},[415],[394,9378],{"className":9379,"style":466},[465],[394,9381],{"className":9382,"style":626},[465],[394,9384,457],{"className":9385},[516],[394,9387],{"className":9388,"style":466},[465],[394,9390],{"className":9391,"style":626},[465],[394,9393,9395,9399,9455,9458,9461,9501,9504,9508],{"className":9394},[406],[394,9396],{"className":9397,"style":9398},[410],"height:2.4638em;vertical-align:-1.4138em;",[394,9400,9402],{"className":9401},[4314,4315],[394,9403,9405,9446],{"className":9404},[427,913],[394,9406,9408,9443],{"className":9407},[431],[394,9409,9411,9433],{"className":9410,"style":9295},[435],[394,9412,9413,9416],{"style":9127},[394,9414],{"className":9415,"style":4332},[442],[394,9417,9419],{"className":9418},[447,448,449,450],[394,9420,9422,9425,9428],{"className":9421},[415,450],[394,9423,5212],{"className":9424},[415,419,450],[394,9426,2053],{"className":9427},[470,450],[394,9429,9432],{"className":9430,"style":9431},[415,419,450],"margin-right:0.0572em;","j",[394,9434,9435,9438],{"style":4350},[394,9436],{"className":9437,"style":4332},[442],[394,9439,9440],{},[394,9441,4361],{"className":9442},[4314,4359,4360],[394,9444,983],{"className":9445},[982],[394,9447,9449],{"className":9448},[431],[394,9450,9453],{"className":9451,"style":9452},[435],"height:1.4138em;",[394,9454],{},[394,9456],{"className":9457,"style":734},[465],[394,9459,5640],{"className":9460},[415],[394,9462,9464,9467],{"className":9463},[415],[394,9465,9192],{"className":9466},[415,419],[394,9468,9470],{"className":9469},[423],[394,9471,9473,9493],{"className":9472},[427,913],[394,9474,9476,9490],{"className":9475},[431],[394,9477,9479],{"className":9478,"style":5200},[435],[394,9480,9481,9484],{"style":5016},[394,9482],{"className":9483,"style":443},[442],[394,9485,9487],{"className":9486},[447,448,449,450],[394,9488,5212],{"className":9489},[415,419,450],[394,9491,983],{"className":9492},[982],[394,9494,9496],{"className":9495},[431],[394,9497,9499],{"className":9498,"style":5035},[435],[394,9500],{},[394,9502],{"className":9503,"style":626},[465],[394,9505,9507],{"className":9506},[516],"∩",[394,9509],{"className":9510,"style":626},[465],[394,9512,9514,9518,9559,9562,9565,9568,9571,9574],{"className":9513},[406],[394,9515],{"className":9516,"style":9517},[410],"height:1.0361em;vertical-align:-0.2861em;",[394,9519,9521,9524],{"className":9520},[415],[394,9522,9192],{"className":9523},[415,419],[394,9525,9527],{"className":9526},[423],[394,9528,9530,9550],{"className":9529},[427,913],[394,9531,9533,9547],{"className":9532},[431],[394,9534,9536],{"className":9535,"style":5200},[435],[394,9537,9538,9541],{"style":5016},[394,9539],{"className":9540,"style":443},[442],[394,9542,9544],{"className":9543},[447,448,449,450],[394,9545,9432],{"className":9546,"style":9431},[415,419,450],[394,9548,983],{"className":9549},[982],[394,9551,9553],{"className":9552},[431],[394,9554,9557],{"className":9555,"style":9556},[435],"height:0.2861em;",[394,9558],{},[394,9560,5640],{"className":9561},[415],[394,9563],{"className":9564,"style":466},[465],[394,9566],{"className":9567,"style":626},[465],[394,9569,1080],{"className":9570},[516],[394,9572],{"className":9573,"style":466},[465],[394,9575],{"className":9576,"style":626},[465],[394,9578,9580,9584,9644,9647,9650,9690,9693,9696],{"className":9579},[406],[394,9581],{"className":9582,"style":9583},[410],"height:2.4882em;vertical-align:-1.4382em;",[394,9585,9587],{"className":9586},[4314,4315],[394,9588,9590,9635],{"className":9589},[427,913],[394,9591,9593,9632],{"className":9592},[431],[394,9594,9596,9622],{"className":9595,"style":9295},[435],[394,9597,9598,9601],{"style":4328},[394,9599],{"className":9600,"style":4332},[442],[394,9602,9604],{"className":9603},[447,448,449,450],[394,9605,9607,9610,9613,9616,9619],{"className":9606},[415,450],[394,9608,5212],{"className":9609},[415,419,450],[394,9611,2053],{"className":9612},[470,450],[394,9614,9432],{"className":9615,"style":9431},[415,419,450],[394,9617,2053],{"className":9618},[470,450],[394,9620,2068],{"className":9621,"style":2067},[415,419,450],[394,9623,9624,9627],{"style":4350},[394,9625],{"className":9626,"style":4332},[442],[394,9628,9629],{},[394,9630,4361],{"className":9631},[4314,4359,4360],[394,9633,983],{"className":9634},[982],[394,9636,9638],{"className":9637},[431],[394,9639,9642],{"className":9640,"style":9641},[435],"height:1.4382em;",[394,9643],{},[394,9645],{"className":9646,"style":734},[465],[394,9648,5640],{"className":9649},[415],[394,9651,9653,9656],{"className":9652},[415],[394,9654,9192],{"className":9655},[415,419],[394,9657,9659],{"className":9658},[423],[394,9660,9662,9682],{"className":9661},[427,913],[394,9663,9665,9679],{"className":9664},[431],[394,9666,9668],{"className":9667,"style":5200},[435],[394,9669,9670,9673],{"style":5016},[394,9671],{"className":9672,"style":443},[442],[394,9674,9676],{"className":9675},[447,448,449,450],[394,9677,5212],{"className":9678},[415,419,450],[394,9680,983],{"className":9681},[982],[394,9683,9685],{"className":9684},[431],[394,9686,9688],{"className":9687,"style":5035},[435],[394,9689],{},[394,9691],{"className":9692,"style":626},[465],[394,9694,9507],{"className":9695},[516],[394,9697],{"className":9698,"style":626},[465],[394,9700,9702,9706,9746,9749,9752],{"className":9701},[406],[394,9703],{"className":9704,"style":9705},[410],"height:0.9694em;vertical-align:-0.2861em;",[394,9707,9709,9712],{"className":9708},[415],[394,9710,9192],{"className":9711},[415,419],[394,9713,9715],{"className":9714},[423],[394,9716,9718,9738],{"className":9717},[427,913],[394,9719,9721,9735],{"className":9720},[431],[394,9722,9724],{"className":9723,"style":5200},[435],[394,9725,9726,9729],{"style":5016},[394,9727],{"className":9728,"style":443},[442],[394,9730,9732],{"className":9731},[447,448,449,450],[394,9733,9432],{"className":9734,"style":9431},[415,419,450],[394,9736,983],{"className":9737},[982],[394,9739,9741],{"className":9740},[431],[394,9742,9744],{"className":9743,"style":9556},[435],[394,9745],{},[394,9747],{"className":9748,"style":626},[465],[394,9750,9507],{"className":9751},[516],[394,9753],{"className":9754,"style":626},[465],[394,9756,9758,9761,9801,9804,9807,9810,9813,9816],{"className":9757},[406],[394,9759],{"className":9760,"style":535},[410],[394,9762,9764,9767],{"className":9763},[415],[394,9765,9192],{"className":9766},[415,419],[394,9768,9770],{"className":9769},[423],[394,9771,9773,9793],{"className":9772},[427,913],[394,9774,9776,9790],{"className":9775},[431],[394,9777,9779],{"className":9778,"style":5145},[435],[394,9780,9781,9784],{"style":5016},[394,9782],{"className":9783,"style":443},[442],[394,9785,9787],{"className":9786},[447,448,449,450],[394,9788,2068],{"className":9789,"style":2067},[415,419,450],[394,9791,983],{"className":9792},[982],[394,9794,9796],{"className":9795},[431],[394,9797,9799],{"className":9798,"style":5035},[435],[394,9800],{},[394,9802,5640],{"className":9803},[415],[394,9805],{"className":9806,"style":466},[465],[394,9808],{"className":9809,"style":626},[465],[394,9811,457],{"className":9812},[516],[394,9814],{"className":9815,"style":466},[465],[394,9817],{"className":9818,"style":626},[465],[394,9820,9822,9825,9828,9831,9834,9837,9840],{"className":9821},[406],[394,9823],{"className":9824,"style":619},[410],[394,9826,739],{"className":9827},[738],[394,9829],{"className":9830,"style":466},[465],[394,9832],{"className":9833,"style":626},[465],[394,9835,1080],{"className":9836},[516],[394,9838],{"className":9839,"style":466},[465],[394,9841],{"className":9842,"style":626},[465],[394,9844,9846,9850,9853,9856,9859,9898,9935,9980,9983,10023,10060],{"className":9845},[406],[394,9847],{"className":9848,"style":9849},[410],"height:2.4277em;vertical-align:-1.2777em;",[394,9851,540],{"className":9852},[539],[394,9854,457],{"className":9855},[415],[394,9857,461],{"className":9858},[415],[394,9860,9862,9865],{"className":9861},[562],[394,9863,563],{"className":9864},[562],[394,9866,9868],{"className":9867},[423],[394,9869,9871],{"className":9870},[427],[394,9872,9874],{"className":9873},[431],[394,9875,9878],{"className":9876,"style":9877},[435],"height:0.8641em;",[394,9879,9880,9883],{"style":4285},[394,9881],{"className":9882,"style":443},[442],[394,9884,9886],{"className":9885},[447,448,449,450],[394,9887,9889,9892,9895],{"className":9888},[415,450],[394,9890,605],{"className":9891},[415,419,450],[394,9893,1080],{"className":9894},[516,450],[394,9896,461],{"className":9897},[415,450],[394,9899,9901],{"className":9900},[539],[394,9902,9904],{"className":9903},[1230,9069],[394,9905,9907,9927],{"className":9906},[427,913],[394,9908,9910,9924],{"className":9909},[431],[394,9911,9913],{"className":9912,"style":9079},[435],[394,9914,9915,9918],{"style":9082},[394,9916],{"className":9917,"style":9086},[442],[394,9919,9920],{"style":9089},[2823,9921,9922],{"xmlns":2825,"width":9092,"height":9093,"viewBox":9094},[2835,9923],{"d":9097},[394,9925,983],{"className":9926},[982],[394,9928,9930],{"className":9929},[431],[394,9931,9933],{"className":9932,"style":9107},[435],[394,9934],{},[394,9936,9938],{"className":9937},[4314,4315],[394,9939,9941,9972],{"className":9940},[427,913],[394,9942,9944,9969],{"className":9943},[431],[394,9945,9947,9958],{"className":9946,"style":9295},[435],[394,9948,9949,9952],{"style":9127},[394,9950],{"className":9951,"style":4332},[442],[394,9953,9955],{"className":9954},[447,448,449,450],[394,9956,5212],{"className":9957},[415,419,450],[394,9959,9960,9963],{"style":4350},[394,9961],{"className":9962,"style":4332},[442],[394,9964,9965],{},[394,9966,9968],{"className":9967},[4314,4359,4360],"⋂",[394,9970,983],{"className":9971},[982],[394,9973,9975],{"className":9974},[431],[394,9976,9978],{"className":9977,"style":9180},[435],[394,9979],{},[394,9981],{"className":9982,"style":734},[465],[394,9984,9986,9989],{"className":9985},[415],[394,9987,9192],{"className":9988},[415,419],[394,9990,9992],{"className":9991},[423],[394,9993,9995,10015],{"className":9994},[427,913],[394,9996,9998,10012],{"className":9997},[431],[394,9999,10001],{"className":10000,"style":5200},[435],[394,10002,10003,10006],{"style":5016},[394,10004],{"className":10005,"style":443},[442],[394,10007,10009],{"className":10008},[447,448,449,450],[394,10010,5212],{"className":10011},[415,419,450],[394,10013,983],{"className":10014},[982],[394,10016,10018],{"className":10017},[431],[394,10019,10021],{"className":10020,"style":5035},[435],[394,10022],{},[394,10024,10026],{"className":10025},[562],[394,10027,10029],{"className":10028},[1230,9069],[394,10030,10032,10052],{"className":10031},[427,913],[394,10033,10035,10049],{"className":10034},[431],[394,10036,10038],{"className":10037,"style":9079},[435],[394,10039,10040,10043],{"style":9082},[394,10041],{"className":10042,"style":9086},[442],[394,10044,10045],{"style":9089},[2823,10046,10047],{"xmlns":2825,"width":9092,"height":9093,"viewBox":9094},[2835,10048],{"d":9097},[394,10050,983],{"className":10051},[982],[394,10053,10055],{"className":10054},[431],[394,10056,10058],{"className":10057,"style":9107},[435],[394,10059],{},[394,10061,1099],{"className":10062},[415],[2022,10064,10065],{"type":2024},[381,10066,10067,10070,10071,10086,10087,10298,10299],{},[389,10068,10069],{},"Lemma (Alternating-sign principle)."," Add the sizes of all single sets, subtract all\npairwise intersections, add all triple intersections, and so on. An element lying\nin exactly ",[394,10072,10074],{"className":10073},[397],[394,10075,10077],{"className":10076,"ariaHidden":402},[401],[394,10078,10080,10083],{"className":10079},[406],[394,10081],{"className":10082,"style":601},[410],[394,10084,3715],{"className":10085},[415,419]," of the sets is counted ",[394,10088,10090],{"className":10089},[397],[394,10091,10093,10289],{"className":10092,"ariaHidden":402},[401],[394,10094,10096,10100,10166,10169,10172,10175,10213,10280,10283,10286],{"className":10095},[406],[394,10097],{"className":10098,"style":10099},[410],"height:1.3311em;vertical-align:-0.4811em;",[394,10101,10103,10106],{"className":10102},[4314],[394,10104,4361],{"className":10105,"style":4752},[4314,4359,4751],[394,10107,10109],{"className":10108},[423],[394,10110,10112,10157],{"className":10111},[427,913],[394,10113,10115,10154],{"className":10114},[431],[394,10116,10119,10139],{"className":10117,"style":10118},[435],"height:0.8043em;",[394,10120,10121,10124],{"style":4768},[394,10122],{"className":10123,"style":443},[442],[394,10125,10127],{"className":10126},[447,448,449,450],[394,10128,10130,10133,10136],{"className":10129},[415,450],[394,10131,9432],{"className":10132,"style":9431},[415,419,450],[394,10134,674],{"className":10135},[470,450],[394,10137,461],{"className":10138},[415,450],[394,10140,10142,10145],{"style":10141},"top:-3.2029em;margin-right:0.05em;",[394,10143],{"className":10144,"style":443},[442],[394,10146,10148],{"className":10147},[447,448,449,450],[394,10149,10151],{"className":10150},[415,450],[394,10152,3715],{"className":10153},[415,419,450],[394,10155,983],{"className":10156},[982],[394,10158,10160],{"className":10159},[431],[394,10161,10164],{"className":10162,"style":10163},[435],"height:0.4358em;",[394,10165],{},[394,10167,540],{"className":10168},[539],[394,10170,457],{"className":10171},[415],[394,10173,461],{"className":10174},[415],[394,10176,10178,10181],{"className":10177},[562],[394,10179,563],{"className":10180},[562],[394,10182,10184],{"className":10183},[423],[394,10185,10187],{"className":10186},[427],[394,10188,10190],{"className":10189},[431],[394,10191,10193],{"className":10192,"style":8576},[435],[394,10194,10195,10198],{"style":438},[394,10196],{"className":10197,"style":443},[442],[394,10199,10201],{"className":10200},[447,448,449,450],[394,10202,10204,10207,10210],{"className":10203},[415,450],[394,10205,9432],{"className":10206,"style":9431},[415,419,450],[394,10208,1080],{"className":10209},[516,450],[394,10211,461],{"className":10212},[415,450],[394,10214,10216,10222,10274],{"className":10215},[415],[394,10217,10219],{"className":10218,"style":1226},[539,1225],[394,10220,540],{"className":10221},[1230,1507],[394,10223,10225],{"className":10224},[909],[394,10226,10228,10265],{"className":10227},[427,913],[394,10229,10231,10262],{"className":10230},[431],[394,10232,10234,10248],{"className":10233,"style":1520},[435],[394,10235,10236,10239],{"style":1523},[394,10237],{"className":10238,"style":443},[442],[394,10240,10242],{"className":10241},[447,448,449,450],[394,10243,10245],{"className":10244},[415,450],[394,10246,9432],{"className":10247,"style":9431},[415,419,450],[394,10249,10250,10253],{"style":1538},[394,10251],{"className":10252,"style":443},[442],[394,10254,10256],{"className":10255},[447,448,449,450],[394,10257,10259],{"className":10258},[415,450],[394,10260,3715],{"className":10261},[415,419,450],[394,10263,983],{"className":10264},[982],[394,10266,10268],{"className":10267},[431],[394,10269,10272],{"className":10270,"style":10271},[435],"height:0.4811em;",[394,10273],{},[394,10275,10277],{"className":10276,"style":1226},[562,1225],[394,10278,563],{"className":10279},[1230,1507],[394,10281],{"className":10282,"style":466},[465],[394,10284,674],{"className":10285},[470],[394,10287],{"className":10288,"style":466},[465],[394,10290,10292,10295],{"className":10291},[406],[394,10293],{"className":10294,"style":639},[410],[394,10296,461],{"className":10297},[415],"\ntime in total, exactly once, as it should be.",[1609,10300,10301],{},[384,10302,3386],{"href":10303,"ariaDescribedBy":10304,"dataFootnoteRef":376,"id":10305},"#user-content-fn-clrs-ie",[1615],"user-content-fnref-clrs-ie",[2817,10307,10309,10466],{"className":10308},[2820,2821],[2823,10310,10314],{"xmlns":2825,"width":10311,"height":10312,"viewBox":10313},"279.623","224.260","-75 -75 209.717 168.195",[2830,10315,10316,10319,10322,10325,10328,10331,10384,10391,10398,10405,10419,10433,10447,10454,10460],{"stroke":2832,"style":2833},[2835,10317],{"fill":2837,"d":10318,"style":2986},"M20.155 23.764c0-23.571-19.108-42.68-42.68-42.68-23.57 0-42.678 19.109-42.678 42.68s19.107 42.679 42.679 42.679c23.571 0 42.679-19.108 42.679-42.68Zm-42.68 0",[2835,10320],{"fill":2837,"d":10321,"style":2986},"M77.061 23.764c0-23.571-19.108-42.68-42.68-42.68S-8.298.194-8.298 23.765s19.108 42.679 42.68 42.679c23.57 0 42.678-19.108 42.678-42.68Zm-42.68 0",[2835,10323],{"fill":2837,"d":10324,"style":2986},"M48.608-24.606c0-23.571-19.108-42.679-42.68-42.679-23.57 0-42.679 19.108-42.679 42.68 0 23.57 19.108 42.678 42.68 42.678 23.57 0 42.679-19.107 42.679-42.679Zm-42.68 0",[2835,10326],{"fill":2953,"stroke":2837,"d":10327},"M7.229 7.546a1.3 1.3 0 1 0-2.6 0 1.3 1.3 0 0 0 2.6 0m-1.3 0",[2835,10329],{"fill":2837,"stroke":2953,"d":10330},"M5.929 7.546c48.37 33.29 48.37 33.29 62.595 73.123",[2830,10332,10333],{"fill":2953,"stroke":2953},[2830,10334,10335,10342,10348,10354,10360,10366,10372,10378],{"fill":2953,"stroke":2837,"fontSize":7134},[2830,10336,10338],{"transform":10337},"translate(94.582 60.578)",[2835,10339],{"d":10340,"fill":2953,"stroke":2953,"className":10341,"style":2847},"M-19.123 24.441L-19.123 21.712L-21.830 21.712Q-22.010 21.681-22.010 21.514Q-22.010 21.448-21.959 21.393Q-21.909 21.338-21.830 21.325L-19.123 21.325L-19.123 18.596Q-19.109 18.521-19.055 18.475Q-19 18.429-18.925 18.429Q-18.855 18.429-18.802 18.477Q-18.749 18.526-18.736 18.596L-18.736 21.325L-16.024 21.325Q-15.853 21.360-15.853 21.514Q-15.853 21.677-16.024 21.712L-18.736 21.712L-18.736 24.441Q-18.771 24.612-18.925 24.612Q-18.995 24.612-19.052 24.564Q-19.109 24.515-19.123 24.441",[2846],[2830,10343,10344],{"transform":10337},[2835,10345],{"d":10346,"fill":2953,"stroke":2953,"className":10347,"style":2847},"M-12.698 25.843L-12.698 17.185Q-12.667 17.014-12.500 17.014Q-12.426 17.014-12.375 17.062Q-12.325 17.111-12.311 17.185L-12.311 25.843Q-12.325 25.917-12.377 25.966Q-12.430 26.014-12.500 26.014Q-12.667 26.014-12.698 25.843",[2846],[2830,10349,10350],{"transform":10337},[2835,10351],{"d":10352,"fill":2953,"stroke":2953,"className":10353,"style":2847},"M-9.057 23.764L-10.788 23.764Q-10.885 23.764-10.885 23.645Q-10.885 23.588-10.854 23.518Q-10.823 23.448-10.762 23.448Q-10.072 23.448-9.672 22.837Q-9.672 22.837-9.646 22.810L-6.376 17.427Q-6.315 17.322-6.187 17.322L-6.099 17.322Q-5.976 17.322-5.963 17.427L-5.335 23.259Q-5.291 23.377-5.100 23.412Q-4.908 23.448-4.649 23.448Q-4.605 23.448-4.572 23.485Q-4.539 23.522-4.539 23.557Q-4.539 23.764-4.702 23.764L-6.934 23.764Q-6.974 23.764-7.005 23.724Q-7.035 23.685-7.035 23.645Q-7.035 23.584-7 23.516Q-6.965 23.448-6.908 23.448Q-6.631 23.448-6.422 23.404Q-6.214 23.360-6.170 23.206L-6.332 21.721L-8.653 21.721L-9.373 22.916Q-9.457 23.039-9.457 23.162Q-9.457 23.320-9.316 23.384Q-9.176 23.448-8.995 23.448Q-8.951 23.448-8.925 23.481Q-8.899 23.514-8.899 23.557Q-8.899 23.764-9.057 23.764M-6.684 18.482L-8.455 21.404L-6.367 21.404",[2846],[2830,10355,10356],{"transform":10337},[2835,10357],{"d":10358,"fill":2953,"stroke":2953,"className":10359,"style":2847},"M-1.713 23.790L-1.713 20.270Q-1.713 19.655-1.326 19.224Q-0.939 18.794-0.342 18.578Q0.256 18.363 0.849 18.363Q1.306 18.363 1.750 18.480Q2.194 18.596 2.572 18.833Q2.950 19.071 3.183 19.431Q3.416 19.791 3.416 20.270L3.416 23.790Q3.402 23.861 3.347 23.911Q3.293 23.962 3.227 23.962Q3.060 23.962 3.029 23.790L3.029 20.310Q3.029 19.787 2.697 19.438Q2.365 19.088 1.864 18.919Q1.363 18.750 0.849 18.750Q0.339 18.750-0.159 18.919Q-0.658 19.088-0.992 19.438Q-1.326 19.787-1.326 20.310L-1.326 23.790Q-1.339 23.865-1.392 23.913Q-1.445 23.962-1.515 23.962Q-1.682 23.962-1.713 23.790",[2846],[2830,10361,10362],{"transform":10337},[2835,10363],{"d":10364,"fill":2953,"stroke":2953,"className":10365,"style":2847},"M9.972 23.764L6.500 23.764Q6.391 23.764 6.391 23.645Q6.391 23.584 6.423 23.516Q6.456 23.448 6.518 23.448Q7.019 23.448 7.247 23.395Q7.375 23.347 7.445 23.118L8.667 18.201Q8.684 18.113 8.684 18.069Q8.684 18.003 8.658 17.985Q8.487 17.932 7.955 17.932Q7.849 17.932 7.849 17.814Q7.849 17.752 7.882 17.684Q7.915 17.616 7.977 17.616L11.251 17.616Q11.655 17.616 12.048 17.752Q12.442 17.889 12.697 18.172Q12.952 18.455 12.952 18.868Q12.952 19.304 12.657 19.664Q12.363 20.024 11.925 20.248Q11.488 20.472 11.053 20.552Q11.413 20.587 11.741 20.745Q12.068 20.903 12.273 21.180Q12.477 21.457 12.477 21.822Q12.477 22.235 12.246 22.595Q12.016 22.955 11.629 23.217Q11.242 23.478 10.807 23.621Q10.372 23.764 9.972 23.764M8.157 23.386Q8.157 23.448 8.443 23.448L9.792 23.448Q10.240 23.448 10.660 23.210Q11.079 22.973 11.332 22.580Q11.585 22.186 11.585 21.738Q11.585 21.444 11.464 21.206Q11.343 20.969 11.126 20.833Q10.908 20.697 10.614 20.697L8.812 20.697L8.192 23.180Q8.157 23.320 8.157 23.386M9.414 18.266L8.873 20.433L10.288 20.433Q10.706 20.433 11.130 20.220Q11.554 20.007 11.820 19.642Q12.086 19.277 12.086 18.842Q12.086 18.447 11.829 18.190Q11.572 17.932 11.172 17.932L9.884 17.932Q9.625 17.932 9.550 17.979Q9.475 18.025 9.414 18.266",[2846],[2830,10367,10368],{"transform":10337},[2835,10369],{"d":10370,"fill":2953,"stroke":2953,"className":10371,"style":2847},"M16.012 23.790L16.012 20.270Q16.012 19.655 16.399 19.224Q16.786 18.794 17.383 18.578Q17.981 18.363 18.574 18.363Q19.031 18.363 19.475 18.480Q19.919 18.596 20.297 18.833Q20.675 19.071 20.908 19.431Q21.141 19.791 21.141 20.270L21.141 23.790Q21.127 23.861 21.072 23.911Q21.018 23.962 20.952 23.962Q20.785 23.962 20.754 23.790L20.754 20.310Q20.754 19.787 20.422 19.438Q20.090 19.088 19.589 18.919Q19.088 18.750 18.574 18.750Q18.064 18.750 17.566 18.919Q17.067 19.088 16.733 19.438Q16.399 19.787 16.399 20.310L16.399 23.790Q16.386 23.865 16.333 23.913Q16.280 23.962 16.210 23.962Q16.043 23.962 16.012 23.790",[2846],[2830,10373,10374],{"transform":10337},[2835,10375],{"d":10376,"fill":2953,"stroke":2953,"className":10377,"style":2847},"M25.052 21.791Q25.052 22.621 25.539 23.133Q26.027 23.645 26.862 23.645Q27.433 23.645 27.967 23.364Q28.501 23.083 28.886 22.602Q29.270 22.120 29.415 21.567Q29.424 21.536 29.448 21.512Q29.472 21.488 29.508 21.488L29.613 21.488Q29.705 21.488 29.705 21.602Q29.543 22.239 29.090 22.784Q28.637 23.329 28.011 23.645Q27.385 23.962 26.717 23.962Q25.988 23.962 25.412 23.645Q24.836 23.329 24.507 22.764Q24.177 22.200 24.177 21.470Q24.177 20.705 24.526 19.969Q24.876 19.233 25.458 18.664Q26.040 18.095 26.792 17.757Q27.543 17.418 28.299 17.418Q28.769 17.418 29.167 17.623Q29.565 17.827 29.811 18.209L30.523 17.436Q30.540 17.418 30.580 17.418L30.633 17.418Q30.720 17.418 30.720 17.537L30.118 19.932Q30.101 20.011 30.031 20.011L29.894 20.011Q29.802 20.011 29.802 19.892Q29.842 19.712 29.842 19.462Q29.842 19 29.677 18.605Q29.512 18.209 29.182 17.972Q28.853 17.735 28.383 17.735Q27.644 17.735 27.027 18.093Q26.409 18.451 25.968 19.046Q25.526 19.642 25.289 20.365Q25.052 21.088 25.052 21.791",[2846],[2830,10379,10380],{"transform":10337},[2835,10381],{"d":10382,"fill":2953,"stroke":2953,"className":10383,"style":2847},"M32.053 25.843L32.053 17.185Q32.084 17.014 32.251 17.014Q32.325 17.014 32.376 17.062Q32.426 17.111 32.440 17.185L32.440 25.843Q32.426 25.917 32.374 25.966Q32.321 26.014 32.251 26.014Q32.084 26.014 32.053 25.843",[2846],[2830,10385,10387],{"transform":10386},"translate(-33.343 35.795)",[2835,10388],{"d":10389,"fill":2832,"stroke":2832,"className":10390,"style":2847},"M-20.362 23.764L-22.093 23.764Q-22.190 23.764-22.190 23.645Q-22.190 23.588-22.159 23.518Q-22.128 23.448-22.067 23.448Q-21.377 23.448-20.977 22.837Q-20.977 22.837-20.951 22.810L-17.681 17.427Q-17.620 17.322-17.492 17.322L-17.404 17.322Q-17.281 17.322-17.268 17.427L-16.640 23.259Q-16.596 23.377-16.405 23.412Q-16.213 23.448-15.954 23.448Q-15.910 23.448-15.877 23.485Q-15.844 23.522-15.844 23.557Q-15.844 23.764-16.007 23.764L-18.239 23.764Q-18.279 23.764-18.310 23.724Q-18.340 23.685-18.340 23.645Q-18.340 23.584-18.305 23.516Q-18.270 23.448-18.213 23.448Q-17.936 23.448-17.727 23.404Q-17.519 23.360-17.475 23.206L-17.637 21.721L-19.958 21.721L-20.678 22.916Q-20.762 23.039-20.762 23.162Q-20.762 23.320-20.621 23.384Q-20.481 23.448-20.300 23.448Q-20.256 23.448-20.230 23.481Q-20.204 23.514-20.204 23.557Q-20.204 23.764-20.362 23.764M-17.989 18.482L-19.760 21.404L-17.672 21.404",[2846],[2830,10392,10394],{"transform":10393},"translate(83.057 35.795)",[2835,10395],{"d":10396,"fill":2832,"stroke":2832,"className":10397,"style":2847},"M-18.547 23.764L-22.019 23.764Q-22.128 23.764-22.128 23.645Q-22.128 23.584-22.096 23.516Q-22.063 23.448-22.001 23.448Q-21.500 23.448-21.272 23.395Q-21.144 23.347-21.074 23.118L-19.852 18.201Q-19.835 18.113-19.835 18.069Q-19.835 18.003-19.861 17.985Q-20.032 17.932-20.564 17.932Q-20.670 17.932-20.670 17.814Q-20.670 17.752-20.637 17.684Q-20.604 17.616-20.542 17.616L-17.268 17.616Q-16.864 17.616-16.471 17.752Q-16.077 17.889-15.822 18.172Q-15.567 18.455-15.567 18.868Q-15.567 19.304-15.862 19.664Q-16.156 20.024-16.594 20.248Q-17.031 20.472-17.466 20.552Q-17.106 20.587-16.778 20.745Q-16.451 20.903-16.246 21.180Q-16.042 21.457-16.042 21.822Q-16.042 22.235-16.273 22.595Q-16.503 22.955-16.890 23.217Q-17.277 23.478-17.712 23.621Q-18.147 23.764-18.547 23.764M-20.362 23.386Q-20.362 23.448-20.076 23.448L-18.727 23.448Q-18.279 23.448-17.859 23.210Q-17.440 22.973-17.187 22.580Q-16.934 22.186-16.934 21.738Q-16.934 21.444-17.055 21.206Q-17.176 20.969-17.393 20.833Q-17.611 20.697-17.905 20.697L-19.707 20.697L-20.327 23.180Q-20.362 23.320-20.362 23.386M-19.105 18.266L-19.646 20.433L-18.231 20.433Q-17.813 20.433-17.389 20.220Q-16.965 20.007-16.699 19.642Q-16.433 19.277-16.433 18.842Q-16.433 18.447-16.690 18.190Q-16.947 17.932-17.347 17.932L-18.635 17.932Q-18.894 17.932-18.969 17.979Q-19.044 18.025-19.105 18.266",[2846],[2830,10399,10401],{"transform":10400},"translate(24.831 -86.551)",[2835,10402],{"d":10403,"fill":2832,"stroke":2832,"className":10404,"style":2847},"M-21.192 21.791Q-21.192 22.621-20.705 23.133Q-20.217 23.645-19.382 23.645Q-18.811 23.645-18.277 23.364Q-17.743 23.083-17.358 22.602Q-16.974 22.120-16.829 21.567Q-16.820 21.536-16.796 21.512Q-16.772 21.488-16.736 21.488L-16.631 21.488Q-16.539 21.488-16.539 21.602Q-16.701 22.239-17.154 22.784Q-17.607 23.329-18.233 23.645Q-18.859 23.962-19.527 23.962Q-20.256 23.962-20.832 23.645Q-21.408 23.329-21.737 22.764Q-22.067 22.200-22.067 21.470Q-22.067 20.705-21.718 19.969Q-21.368 19.233-20.786 18.664Q-20.204 18.095-19.452 17.757Q-18.701 17.418-17.945 17.418Q-17.475 17.418-17.077 17.623Q-16.679 17.827-16.433 18.209L-15.721 17.436Q-15.704 17.418-15.664 17.418L-15.611 17.418Q-15.524 17.418-15.524 17.537L-16.126 19.932Q-16.143 20.011-16.213 20.011L-16.350 20.011Q-16.442 20.011-16.442 19.892Q-16.402 19.712-16.402 19.462Q-16.402 19-16.567 18.605Q-16.732 18.209-17.062 17.972Q-17.391 17.735-17.861 17.735Q-18.600 17.735-19.217 18.093Q-19.835 18.451-20.276 19.046Q-20.718 19.642-20.955 20.365Q-21.192 21.088-21.192 21.791",[2846],[2830,10406,10407,10413],{"stroke":2837,"fontSize":7134},[2830,10408,10410],{"transform":10409},"translate(-25.56 6.926)",[2835,10411],{"d":10340,"fill":2832,"stroke":2832,"className":10412,"style":2847},[2846],[2830,10414,10415],{"transform":10409},[2835,10416],{"d":10417,"fill":2832,"stroke":2832,"className":10418,"style":2847},"M-13.168 23.764L-14.899 23.764Q-14.996 23.764-14.996 23.645Q-14.996 23.588-14.965 23.518Q-14.934 23.448-14.873 23.448Q-14.183 23.448-13.783 22.837Q-13.783 22.837-13.757 22.810L-10.487 17.427Q-10.426 17.322-10.298 17.322L-10.210 17.322Q-10.087 17.322-10.074 17.427L-9.446 23.259Q-9.402 23.377-9.211 23.412Q-9.019 23.448-8.760 23.448Q-8.716 23.448-8.683 23.485Q-8.650 23.522-8.650 23.557Q-8.650 23.764-8.813 23.764L-11.045 23.764Q-11.085 23.764-11.116 23.724Q-11.146 23.685-11.146 23.645Q-11.146 23.584-11.111 23.516Q-11.076 23.448-11.019 23.448Q-10.742 23.448-10.533 23.404Q-10.325 23.360-10.281 23.206L-10.443 21.721L-12.764 21.721L-13.484 22.916Q-13.568 23.039-13.568 23.162Q-13.568 23.320-13.427 23.384Q-13.287 23.448-13.106 23.448Q-13.062 23.448-13.036 23.481Q-13.010 23.514-13.010 23.557Q-13.010 23.764-13.168 23.764M-10.795 18.482L-12.566 21.404L-10.478 21.404",[2846],[2830,10420,10421,10427],{"stroke":2837,"fontSize":7134},[2830,10422,10424],{"transform":10423},"translate(68.079 6.926)",[2835,10425],{"d":10340,"fill":2832,"stroke":2832,"className":10426,"style":2847},[2846],[2830,10428,10429],{"transform":10423},[2835,10430],{"d":10431,"fill":2832,"stroke":2832,"className":10432,"style":2847},"M-11.353 23.764L-14.825 23.764Q-14.934 23.764-14.934 23.645Q-14.934 23.584-14.902 23.516Q-14.869 23.448-14.807 23.448Q-14.306 23.448-14.078 23.395Q-13.950 23.347-13.880 23.118L-12.658 18.201Q-12.641 18.113-12.641 18.069Q-12.641 18.003-12.667 17.985Q-12.838 17.932-13.370 17.932Q-13.476 17.932-13.476 17.814Q-13.476 17.752-13.443 17.684Q-13.410 17.616-13.348 17.616L-10.074 17.616Q-9.670 17.616-9.277 17.752Q-8.883 17.889-8.628 18.172Q-8.373 18.455-8.373 18.868Q-8.373 19.304-8.668 19.664Q-8.962 20.024-9.400 20.248Q-9.837 20.472-10.272 20.552Q-9.912 20.587-9.584 20.745Q-9.257 20.903-9.052 21.180Q-8.848 21.457-8.848 21.822Q-8.848 22.235-9.079 22.595Q-9.309 22.955-9.696 23.217Q-10.083 23.478-10.518 23.621Q-10.953 23.764-11.353 23.764M-13.168 23.386Q-13.168 23.448-12.882 23.448L-11.533 23.448Q-11.085 23.448-10.665 23.210Q-10.246 22.973-9.993 22.580Q-9.740 22.186-9.740 21.738Q-9.740 21.444-9.861 21.206Q-9.982 20.969-10.199 20.833Q-10.417 20.697-10.711 20.697L-12.513 20.697L-13.133 23.180Q-13.168 23.320-13.168 23.386M-11.911 18.266L-12.452 20.433L-11.037 20.433Q-10.619 20.433-10.195 20.220Q-9.771 20.007-9.505 19.642Q-9.239 19.277-9.239 18.842Q-9.239 18.447-9.496 18.190Q-9.753 17.932-10.153 17.932L-11.441 17.932Q-11.700 17.932-11.775 17.979Q-11.850 18.025-11.911 18.266",[2846],[2830,10434,10435,10441],{"stroke":2837,"fontSize":7134},[2830,10436,10438],{"transform":10437},"translate(21.234 -67.05)",[2835,10439],{"d":10340,"fill":2832,"stroke":2832,"className":10440,"style":2847},[2846],[2830,10442,10443],{"transform":10437},[2835,10444],{"d":10445,"fill":2832,"stroke":2832,"className":10446,"style":2847},"M-13.998 21.791Q-13.998 22.621-13.511 23.133Q-13.023 23.645-12.188 23.645Q-11.617 23.645-11.083 23.364Q-10.549 23.083-10.164 22.602Q-9.780 22.120-9.635 21.567Q-9.626 21.536-9.602 21.512Q-9.578 21.488-9.542 21.488L-9.437 21.488Q-9.345 21.488-9.345 21.602Q-9.507 22.239-9.960 22.784Q-10.413 23.329-11.039 23.645Q-11.665 23.962-12.333 23.962Q-13.062 23.962-13.638 23.645Q-14.214 23.329-14.543 22.764Q-14.873 22.200-14.873 21.470Q-14.873 20.705-14.524 19.969Q-14.174 19.233-13.592 18.664Q-13.010 18.095-12.258 17.757Q-11.507 17.418-10.751 17.418Q-10.281 17.418-9.883 17.623Q-9.485 17.827-9.239 18.209L-8.527 17.436Q-8.510 17.418-8.470 17.418L-8.417 17.418Q-8.330 17.418-8.330 17.537L-8.932 19.932Q-8.949 20.011-9.019 20.011L-9.156 20.011Q-9.248 20.011-9.248 19.892Q-9.208 19.712-9.208 19.462Q-9.208 19-9.373 18.605Q-9.538 18.209-9.868 17.972Q-10.197 17.735-10.667 17.735Q-11.406 17.735-12.023 18.093Q-12.641 18.451-13.082 19.046Q-13.524 19.642-13.761 20.365Q-13.998 21.088-13.998 21.791",[2846],[2830,10448,10450],{"transform":10449},"translate(24.856 6.518)",[2835,10451],{"d":10452,"fill":2832,"stroke":2832,"className":10453,"style":2847},"M-16.288 21.712L-21.570 21.712Q-21.645 21.699-21.698 21.646Q-21.751 21.593-21.751 21.514Q-21.751 21.448-21.696 21.393Q-21.641 21.338-21.570 21.325L-16.288 21.325Q-16.218 21.338-16.167 21.389Q-16.117 21.439-16.117 21.514Q-16.117 21.681-16.288 21.712",[2846],[2830,10455,10457],{"transform":10456},"translate(-5.02 -26.203)",[2835,10458],{"d":10452,"fill":2832,"stroke":2832,"className":10459,"style":2847},[2846],[2830,10461,10463],{"transform":10462},"translate(54.731 -26.203)",[2835,10464],{"d":10452,"fill":2832,"stroke":2832,"className":10465,"style":2847},[2846],[3003,10467,10469],{"className":10468},[3006],"Inclusion–exclusion — add singles, subtract pairs, add the triple (in acc)",[381,10471,10472,10475,10476,10507,10508,10547,10548,10586,10587,10620,10621,10735,10736,10928,10929,11029],{},[389,10473,10474],{},"Worked example (counting coprime-to-a-set integers)."," How many integers in\n",[394,10477,10479],{"className":10478},[397],[394,10480,10482],{"className":10481,"ariaHidden":402},[401],[394,10483,10485,10488,10491,10494,10497,10500,10504],{"className":10484},[406],[394,10486],{"className":10487,"style":535},[410],[394,10489,7472],{"className":10490},[539],[394,10492,461],{"className":10493},[415],[394,10495,769],{"className":10496},[768],[394,10498],{"className":10499,"style":734},[465],[394,10501,10503],{"className":10502},[415],"30",[394,10505,7479],{"className":10506},[562]," are divisible by none of ",[394,10509,10511],{"className":10510},[397],[394,10512,10514],{"className":10513,"ariaHidden":402},[401],[394,10515,10517,10520,10523,10526,10529,10532,10535,10538,10541,10544],{"className":10516},[406],[394,10518],{"className":10519,"style":535},[410],[394,10521,2405],{"className":10522},[539],[394,10524,520],{"className":10525},[415],[394,10527,769],{"className":10528},[768],[394,10530],{"className":10531,"style":734},[465],[394,10533,3386],{"className":10534},[415],[394,10536,769],{"className":10537},[768],[394,10539],{"className":10540,"style":734},[465],[394,10542,5573],{"className":10543},[415],[394,10545,2434],{"className":10546},[562],"? Let ",[394,10549,10551],{"className":10550},[397],[394,10552,10554],{"className":10553,"ariaHidden":402},[401],[394,10555,10557,10561,10564,10567,10570,10575,10578,10581],{"className":10556},[406],[394,10558],{"className":10559,"style":10560},[410],"height:0.8778em;vertical-align:-0.1944em;",[394,10562,9192],{"className":10563},[415,419],[394,10565,769],{"className":10566},[768],[394,10568],{"className":10569,"style":734},[465],[394,10571,10574],{"className":10572,"style":10573},[415,419],"margin-right:0.0502em;","B",[394,10576,769],{"className":10577},[768],[394,10579],{"className":10580,"style":734},[465],[394,10582,10585],{"className":10583,"style":10584},[415,419],"margin-right:0.0715em;","C"," be the multiples of\n",[394,10588,10590],{"className":10589},[397],[394,10591,10593],{"className":10592,"ariaHidden":402},[401],[394,10594,10596,10599,10602,10605,10608,10611,10614,10617],{"className":10595},[406],[394,10597],{"className":10598,"style":5769},[410],[394,10600,520],{"className":10601},[415],[394,10603,769],{"className":10604},[768],[394,10606],{"className":10607,"style":734},[465],[394,10609,3386],{"className":10610},[415],[394,10612,769],{"className":10613},[768],[394,10615],{"className":10616,"style":734},[465],[394,10618,5573],{"className":10619},[415]," respectively. Then ",[394,10622,10624],{"className":10623},[397],[394,10625,10627,10651,10689,10726],{"className":10626,"ariaHidden":402},[401],[394,10628,10630,10633,10636,10639,10642,10645,10648],{"className":10629},[406],[394,10631],{"className":10632,"style":535},[410],[394,10634,5640],{"className":10635},[415],[394,10637,9192],{"className":10638},[415,419],[394,10640,5640],{"className":10641},[415],[394,10643],{"className":10644,"style":466},[465],[394,10646,674],{"className":10647},[470],[394,10649],{"className":10650,"style":466},[465],[394,10652,10654,10657,10661,10664,10668,10671,10674,10677,10680,10683,10686],{"className":10653},[406],[394,10655],{"className":10656,"style":535},[410],[394,10658,10660],{"className":10659},[415],"15",[394,10662,769],{"className":10663},[768],[394,10665,10667],{"className":10666},[465]," ",[394,10669],{"className":10670,"style":734},[465],[394,10672,5640],{"className":10673},[415],[394,10675,10574],{"className":10676,"style":10573},[415,419],[394,10678,5640],{"className":10679},[415],[394,10681],{"className":10682,"style":466},[465],[394,10684,674],{"className":10685},[470],[394,10687],{"className":10688,"style":466},[465],[394,10690,10692,10695,10699,10702,10705,10708,10711,10714,10717,10720,10723],{"className":10691},[406],[394,10693],{"className":10694,"style":535},[410],[394,10696,10698],{"className":10697},[415],"10",[394,10700,769],{"className":10701},[768],[394,10703,10667],{"className":10704},[465],[394,10706],{"className":10707,"style":734},[465],[394,10709,5640],{"className":10710},[415],[394,10712,10585],{"className":10713,"style":10584},[415,419],[394,10715,5640],{"className":10716},[415],[394,10718],{"className":10719,"style":466},[465],[394,10721,674],{"className":10722},[470],[394,10724],{"className":10725,"style":466},[465],[394,10727,10729,10732],{"className":10728},[406],[394,10730],{"className":10731,"style":639},[410],[394,10733,3366],{"className":10734},[415],";\n",[394,10737,10739],{"className":10738},[397],[394,10740,10742,10763,10784,10811,10844,10865,10898,10919],{"className":10741,"ariaHidden":402},[401],[394,10743,10745,10748,10751,10754,10757,10760],{"className":10744},[406],[394,10746],{"className":10747,"style":535},[410],[394,10749,5640],{"className":10750},[415],[394,10752,9192],{"className":10753},[415,419],[394,10755],{"className":10756,"style":626},[465],[394,10758,9507],{"className":10759},[516],[394,10761],{"className":10762,"style":626},[465],[394,10764,10766,10769,10772,10775,10778,10781],{"className":10765},[406],[394,10767],{"className":10768,"style":535},[410],[394,10770,10574],{"className":10771,"style":10573},[415,419],[394,10773,5640],{"className":10774},[415],[394,10776],{"className":10777,"style":466},[465],[394,10779,674],{"className":10780},[470],[394,10782],{"className":10783,"style":466},[465],[394,10785,10787,10790,10794,10798,10802,10805,10808],{"className":10786},[406],[394,10788],{"className":10789,"style":535},[410],[394,10791,10793],{"className":10792},[539],"⌊",[394,10795,10797],{"className":10796},[415],"30\u002F6",[394,10799,10801],{"className":10800},[562],"⌋",[394,10803],{"className":10804,"style":466},[465],[394,10806,674],{"className":10807},[470],[394,10809],{"className":10810,"style":466},[465],[394,10812,10814,10817,10820,10823,10826,10829,10832,10835,10838,10841],{"className":10813},[406],[394,10815],{"className":10816,"style":535},[410],[394,10818,5573],{"className":10819},[415],[394,10821,769],{"className":10822},[768],[394,10824,10667],{"className":10825},[465],[394,10827],{"className":10828,"style":734},[465],[394,10830,5640],{"className":10831},[415],[394,10833,9192],{"className":10834},[415,419],[394,10836],{"className":10837,"style":626},[465],[394,10839,9507],{"className":10840},[516],[394,10842],{"className":10843,"style":626},[465],[394,10845,10847,10850,10853,10856,10859,10862],{"className":10846},[406],[394,10848],{"className":10849,"style":535},[410],[394,10851,10585],{"className":10852,"style":10584},[415,419],[394,10854,5640],{"className":10855},[415],[394,10857],{"className":10858,"style":466},[465],[394,10860,674],{"className":10861},[470],[394,10863],{"className":10864,"style":466},[465],[394,10866,10868,10871,10874,10877,10880,10883,10886,10889,10892,10895],{"className":10867},[406],[394,10869],{"className":10870,"style":535},[410],[394,10872,3386],{"className":10873},[415],[394,10875,769],{"className":10876},[768],[394,10878,10667],{"className":10879},[465],[394,10881],{"className":10882,"style":734},[465],[394,10884,5640],{"className":10885},[415],[394,10887,10574],{"className":10888,"style":10573},[415,419],[394,10890],{"className":10891,"style":626},[465],[394,10893,9507],{"className":10894},[516],[394,10896],{"className":10897,"style":626},[465],[394,10899,10901,10904,10907,10910,10913,10916],{"className":10900},[406],[394,10902],{"className":10903,"style":535},[410],[394,10905,10585],{"className":10906,"style":10584},[415,419],[394,10908,5640],{"className":10909},[415],[394,10911],{"className":10912,"style":466},[465],[394,10914,674],{"className":10915},[470],[394,10917],{"className":10918,"style":466},[465],[394,10920,10922,10925],{"className":10921},[406],[394,10923],{"className":10924,"style":639},[410],[394,10926,520],{"className":10927},[415],"; and\n",[394,10930,10932],{"className":10931},[397],[394,10933,10935,10956,10974,10995,11020],{"className":10934,"ariaHidden":402},[401],[394,10936,10938,10941,10944,10947,10950,10953],{"className":10937},[406],[394,10939],{"className":10940,"style":535},[410],[394,10942,5640],{"className":10943},[415],[394,10945,9192],{"className":10946},[415,419],[394,10948],{"className":10949,"style":626},[465],[394,10951,9507],{"className":10952},[516],[394,10954],{"className":10955,"style":626},[465],[394,10957,10959,10962,10965,10968,10971],{"className":10958},[406],[394,10960],{"className":10961,"style":871},[410],[394,10963,10574],{"className":10964,"style":10573},[415,419],[394,10966],{"className":10967,"style":626},[465],[394,10969,9507],{"className":10970},[516],[394,10972],{"className":10973,"style":626},[465],[394,10975,10977,10980,10983,10986,10989,10992],{"className":10976},[406],[394,10978],{"className":10979,"style":535},[410],[394,10981,10585],{"className":10982,"style":10584},[415,419],[394,10984,5640],{"className":10985},[415],[394,10987],{"className":10988,"style":466},[465],[394,10990,674],{"className":10991},[470],[394,10993],{"className":10994,"style":466},[465],[394,10996,10998,11001,11004,11008,11011,11014,11017],{"className":10997},[406],[394,10999],{"className":11000,"style":535},[410],[394,11002,10793],{"className":11003},[539],[394,11005,11007],{"className":11006},[415],"30\u002F30",[394,11009,10801],{"className":11010},[562],[394,11012],{"className":11013,"style":466},[465],[394,11015,674],{"className":11016},[470],[394,11018],{"className":11019,"style":466},[465],[394,11021,11023,11026],{"className":11022},[406],[394,11024],{"className":11025,"style":639},[410],[394,11027,461],{"className":11028},[415],". So",[394,11031,11033],{"className":11032},[647],[394,11034,11036],{"className":11035},[397],[394,11037,11039,11061,11079,11100,11121,11139,11160,11181,11199,11220,11238],{"className":11038,"ariaHidden":402},[401],[394,11040,11042,11045,11048,11051,11054,11058],{"className":11041},[406],[394,11043],{"className":11044,"style":535},[410],[394,11046,5640],{"className":11047},[415],[394,11049,9192],{"className":11050},[415,419],[394,11052],{"className":11053,"style":626},[465],[394,11055,11057],{"className":11056},[516],"∪",[394,11059],{"className":11060,"style":626},[465],[394,11062,11064,11067,11070,11073,11076],{"className":11063},[406],[394,11065],{"className":11066,"style":871},[410],[394,11068,10574],{"className":11069,"style":10573},[415,419],[394,11071],{"className":11072,"style":626},[465],[394,11074,11057],{"className":11075},[516],[394,11077],{"className":11078,"style":626},[465],[394,11080,11082,11085,11088,11091,11094,11097],{"className":11081},[406],[394,11083],{"className":11084,"style":535},[410],[394,11086,10585],{"className":11087,"style":10584},[415,419],[394,11089,5640],{"className":11090},[415],[394,11092],{"className":11093,"style":466},[465],[394,11095,674],{"className":11096},[470],[394,11098],{"className":11099,"style":466},[465],[394,11101,11103,11106,11109,11112,11115,11118],{"className":11102},[406],[394,11104],{"className":11105,"style":535},[410],[394,11107,540],{"className":11108},[539],[394,11110,10660],{"className":11111},[415],[394,11113],{"className":11114,"style":626},[465],[394,11116,1080],{"className":11117},[516],[394,11119],{"className":11120,"style":626},[465],[394,11122,11124,11127,11130,11133,11136],{"className":11123},[406],[394,11125],{"className":11126,"style":3382},[410],[394,11128,10698],{"className":11129},[415],[394,11131],{"className":11132,"style":626},[465],[394,11134,1080],{"className":11135},[516],[394,11137],{"className":11138,"style":626},[465],[394,11140,11142,11145,11148,11151,11154,11157],{"className":11141},[406],[394,11143],{"className":11144,"style":535},[410],[394,11146,3366],{"className":11147},[415],[394,11149,563],{"className":11150},[562],[394,11152],{"className":11153,"style":626},[465],[394,11155,457],{"className":11156},[516],[394,11158],{"className":11159,"style":626},[465],[394,11161,11163,11166,11169,11172,11175,11178],{"className":11162},[406],[394,11164],{"className":11165,"style":535},[410],[394,11167,540],{"className":11168},[539],[394,11170,5573],{"className":11171},[415],[394,11173],{"className":11174,"style":626},[465],[394,11176,1080],{"className":11177},[516],[394,11179],{"className":11180,"style":626},[465],[394,11182,11184,11187,11190,11193,11196],{"className":11183},[406],[394,11185],{"className":11186,"style":3382},[410],[394,11188,3386],{"className":11189},[415],[394,11191],{"className":11192,"style":626},[465],[394,11194,1080],{"className":11195},[516],[394,11197],{"className":11198,"style":626},[465],[394,11200,11202,11205,11208,11211,11214,11217],{"className":11201},[406],[394,11203],{"className":11204,"style":535},[410],[394,11206,520],{"className":11207},[415],[394,11209,563],{"className":11210},[562],[394,11212],{"className":11213,"style":626},[465],[394,11215,1080],{"className":11216},[516],[394,11218],{"className":11219,"style":626},[465],[394,11221,11223,11226,11229,11232,11235],{"className":11222},[406],[394,11224],{"className":11225,"style":639},[410],[394,11227,461],{"className":11228},[415],[394,11230],{"className":11231,"style":466},[465],[394,11233,674],{"className":11234},[470],[394,11236],{"className":11237,"style":466},[465],[394,11239,11241,11244,11248],{"className":11240},[406],[394,11242],{"className":11243,"style":5769},[410],[394,11245,11247],{"className":11246},[415],"22",[394,11249,769],{"className":11250},[768],[381,11252,11253,11254,11305,11306,11339,11340,11424,11425,11478,11479,1105,11498,392,11501,1099],{},"leaving ",[394,11255,11257],{"className":11256},[397],[394,11258,11260,11278,11296],{"className":11259,"ariaHidden":402},[401],[394,11261,11263,11266,11269,11272,11275],{"className":11262},[406],[394,11264],{"className":11265,"style":3382},[410],[394,11267,10503],{"className":11268},[415],[394,11270],{"className":11271,"style":626},[465],[394,11273,457],{"className":11274},[516],[394,11276],{"className":11277,"style":626},[465],[394,11279,11281,11284,11287,11290,11293],{"className":11280},[406],[394,11282],{"className":11283,"style":639},[410],[394,11285,11247],{"className":11286},[415],[394,11288],{"className":11289,"style":466},[465],[394,11291,674],{"className":11292},[470],[394,11294],{"className":11295,"style":466},[465],[394,11297,11299,11302],{"className":11298},[406],[394,11300],{"className":11301,"style":639},[410],[394,11303,4024],{"className":11304},[415]," integers divisible by none of ",[394,11307,11309],{"className":11308},[397],[394,11310,11312],{"className":11311,"ariaHidden":402},[401],[394,11313,11315,11318,11321,11324,11327,11330,11333,11336],{"className":11314},[406],[394,11316],{"className":11317,"style":5769},[410],[394,11319,520],{"className":11320},[415],[394,11322,769],{"className":11323},[768],[394,11325],{"className":11326,"style":734},[465],[394,11328,3386],{"className":11329},[415],[394,11331,769],{"className":11332},[768],[394,11334],{"className":11335,"style":734},[465],[394,11337,5573],{"className":11338},[415],", which are exactly\n",[394,11341,11343],{"className":11342},[397],[394,11344,11346],{"className":11345,"ariaHidden":402},[401],[394,11347,11349,11352,11355,11358,11361,11364,11367,11370,11374,11377,11380,11384,11387,11390,11394,11397,11400,11404,11407,11410,11414,11417,11420],{"className":11348},[406],[394,11350],{"className":11351,"style":5769},[410],[394,11353,461],{"className":11354},[415],[394,11356,769],{"className":11357},[768],[394,11359],{"className":11360,"style":734},[465],[394,11362,6353],{"className":11363},[415],[394,11365,769],{"className":11366},[768],[394,11368],{"className":11369,"style":734},[465],[394,11371,11373],{"className":11372},[415],"11",[394,11375,769],{"className":11376},[768],[394,11378],{"className":11379,"style":734},[465],[394,11381,11383],{"className":11382},[415],"13",[394,11385,769],{"className":11386},[768],[394,11388],{"className":11389,"style":734},[465],[394,11391,11393],{"className":11392},[415],"17",[394,11395,769],{"className":11396},[768],[394,11398],{"className":11399,"style":734},[465],[394,11401,11403],{"className":11402},[415],"19",[394,11405,769],{"className":11406},[768],[394,11408],{"className":11409,"style":734},[465],[394,11411,11413],{"className":11412},[415],"23",[394,11415,769],{"className":11416},[768],[394,11418],{"className":11419,"style":734},[465],[394,11421,11423],{"className":11422},[415],"29",". The same alternating sum, applied with ",[394,11426,11428],{"className":11427},[397],[394,11429,11431],{"className":11430,"ariaHidden":402},[401],[394,11432,11434,11438],{"className":11433},[406],[394,11435],{"className":11436,"style":11437},[410],"height:0.8333em;vertical-align:-0.15em;",[394,11439,11441,11444],{"className":11440},[415],[394,11442,9192],{"className":11443},[415,419],[394,11445,11447],{"className":11446},[423],[394,11448,11450,11470],{"className":11449},[427,913],[394,11451,11453,11467],{"className":11452},[431],[394,11454,11456],{"className":11455,"style":5200},[435],[394,11457,11458,11461],{"style":5016},[394,11459],{"className":11460,"style":443},[442],[394,11462,11464],{"className":11463},[447,448,449,450],[394,11465,5212],{"className":11466},[415,419,450],[394,11468,983],{"className":11469},[982],[394,11471,11473],{"className":11472},[431],[394,11474,11476],{"className":11475,"style":5035},[435],[394,11477],{}," = ",[1571,11480,11481,11482,11497],{},"maps position ",[394,11483,11485],{"className":11484},[397],[394,11486,11488],{"className":11487,"ariaHidden":402},[401],[394,11489,11491,11494],{"className":11490},[406],[394,11492],{"className":11493,"style":5482},[410],[394,11495,5212],{"className":11496},[415,419]," to itself,",[389,11499,11500],{},"derangements",[394,11502,11504],{"className":11503},[397],[394,11505,11507,11565],{"className":11506,"ariaHidden":402},[401],[394,11508,11510,11513,11556,11559,11562],{"className":11509},[406],[394,11511],{"className":11512,"style":11437},[410],[394,11514,11516,11520],{"className":11515},[415],[394,11517,11519],{"className":11518,"style":821},[415,419],"D",[394,11521,11523],{"className":11522},[423],[394,11524,11526,11548],{"className":11525},[427,913],[394,11527,11529,11545],{"className":11528},[431],[394,11530,11533],{"className":11531,"style":11532},[435],"height:0.1514em;",[394,11534,11536,11539],{"style":11535},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[394,11537],{"className":11538,"style":443},[442],[394,11540,11542],{"className":11541},[447,448,449,450],[394,11543,605],{"className":11544},[415,419,450],[394,11546,983],{"className":11547},[982],[394,11549,11551],{"className":11550},[431],[394,11552,11554],{"className":11553,"style":5035},[435],[394,11555],{},[394,11557],{"className":11558,"style":466},[465],[394,11560,674],{"className":11561},[470],[394,11563],{"className":11564,"style":466},[465],[394,11566,11568,11572,11575,11578,11581,11644,11647,11650,11653,11682,11686,11689],{"className":11567},[406],[394,11569],{"className":11570,"style":11571},[410],"height:1.2605em;vertical-align:-0.4358em;",[394,11573,605],{"className":11574},[415,419],[394,11576,667],{"className":11577},[562],[394,11579],{"className":11580,"style":734},[465],[394,11582,11584,11587],{"className":11583},[4314],[394,11585,4361],{"className":11586,"style":4752},[4314,4359,4751],[394,11588,11590],{"className":11589},[423],[394,11591,11593,11636],{"className":11592},[427,913],[394,11594,11596,11633],{"className":11595},[431],[394,11597,11599,11619],{"className":11598,"style":10118},[435],[394,11600,11601,11604],{"style":4768},[394,11602],{"className":11603,"style":443},[442],[394,11605,11607],{"className":11606},[447,448,449,450],[394,11608,11610,11613,11616],{"className":11609},[415,450],[394,11611,9432],{"className":11612,"style":9431},[415,419,450],[394,11614,674],{"className":11615},[470,450],[394,11617,780],{"className":11618},[415,450],[394,11620,11621,11624],{"style":10141},[394,11622],{"className":11623,"style":443},[442],[394,11625,11627],{"className":11626},[447,448,449,450],[394,11628,11630],{"className":11629},[415,450],[394,11631,605],{"className":11632},[415,419,450],[394,11634,983],{"className":11635},[982],[394,11637,11639],{"className":11638},[431],[394,11640,11642],{"className":11641,"style":10163},[435],[394,11643],{},[394,11645,540],{"className":11646},[539],[394,11648,457],{"className":11649},[415],[394,11651,461],{"className":11652},[415],[394,11654,11656,11659],{"className":11655},[562],[394,11657,563],{"className":11658},[562],[394,11660,11662],{"className":11661},[423],[394,11663,11665],{"className":11664},[427],[394,11666,11668],{"className":11667},[431],[394,11669,11671],{"className":11670,"style":8576},[435],[394,11672,11673,11676],{"style":438},[394,11674],{"className":11675,"style":443},[442],[394,11677,11679],{"className":11678},[447,448,449,450],[394,11680,9432],{"className":11681,"style":9431},[415,419,450],[394,11683,11685],{"className":11684},[415],"\u002F",[394,11687,9432],{"className":11688,"style":9431},[415,419],[394,11690,667],{"className":11691},[562],[578,11693,11695],{"id":11694},"the-chinese-remainder-theorem","The Chinese Remainder Theorem",[381,11697,11698,11699,11701,11702,11705],{},"Inclusion–exclusion combines counts; the ",[389,11700,575],{}," (CRT)\ncombines ",[1107,11703,11704],{},"congruences",". Given a system",[394,11707,11709],{"className":11708},[647],[394,11710,11712],{"className":11711},[397],[394,11713,11715,11734,11786,11871,11923,12023,12075],{"className":11714,"ariaHidden":402},[401],[394,11716,11718,11722,11725,11728,11731],{"className":11717},[406],[394,11719],{"className":11720,"style":11721},[410],"height:0.4637em;",[394,11723,4243],{"className":11724},[415,419],[394,11726],{"className":11727,"style":466},[465],[394,11729,471],{"className":11730},[470],[394,11732],{"className":11733,"style":466},[465],[394,11735,11737,11740,11780,11783],{"className":11736},[406],[394,11738],{"className":11739,"style":5126},[410],[394,11741,11743,11746],{"className":11742},[415],[394,11744,384],{"className":11745},[415,419],[394,11747,11749],{"className":11748},[423],[394,11750,11752,11772],{"className":11751},[427,913],[394,11753,11755,11769],{"className":11754},[431],[394,11756,11758],{"className":11757,"style":5013},[435],[394,11759,11760,11763],{"style":5016},[394,11761],{"className":11762,"style":443},[442],[394,11764,11766],{"className":11765},[447,448,449,450],[394,11767,461],{"className":11768},[415,450],[394,11770,983],{"className":11771},[982],[394,11773,11775],{"className":11774},[431],[394,11776,11778],{"className":11777,"style":5035},[435],[394,11779],{},[394,11781],{"className":11782},[465,524],[394,11784],{"className":11785,"style":5694},[465],[394,11787,11789,11792,11795,11804,11807,11847,11850,11853,11856,11859,11862,11865,11868],{"className":11788},[406],[394,11790],{"className":11791,"style":535},[410],[394,11793,540],{"className":11794},[539],[394,11796,11798],{"className":11797},[415],[394,11799,11801],{"className":11800},[415],[394,11802,551],{"className":11803},[415,550],[394,11805],{"className":11806,"style":555},[465],[394,11808,11810,11813],{"className":11809},[415],[394,11811,3715],{"className":11812},[415,419],[394,11814,11816],{"className":11815},[423],[394,11817,11819,11839],{"className":11818},[427,913],[394,11820,11822,11836],{"className":11821},[431],[394,11823,11825],{"className":11824,"style":5013},[435],[394,11826,11827,11830],{"style":5016},[394,11828],{"className":11829,"style":443},[442],[394,11831,11833],{"className":11832},[447,448,449,450],[394,11834,461],{"className":11835},[415,450],[394,11837,983],{"className":11838},[982],[394,11840,11842],{"className":11841},[431],[394,11843,11845],{"className":11844,"style":5035},[435],[394,11846],{},[394,11848,563],{"className":11849},[562],[394,11851,769],{"className":11852},[768],[394,11854],{"className":11855,"style":5694},[465],[394,11857],{"className":11858,"style":734},[465],[394,11860,4243],{"className":11861},[415,419],[394,11863],{"className":11864,"style":466},[465],[394,11866,471],{"className":11867},[470],[394,11869],{"className":11870,"style":466},[465],[394,11872,11874,11877,11917,11920],{"className":11873},[406],[394,11875],{"className":11876,"style":5126},[410],[394,11878,11880,11883],{"className":11879},[415],[394,11881,384],{"className":11882},[415,419],[394,11884,11886],{"className":11885},[423],[394,11887,11889,11909],{"className":11888},[427,913],[394,11890,11892,11906],{"className":11891},[431],[394,11893,11895],{"className":11894,"style":5013},[435],[394,11896,11897,11900],{"style":5016},[394,11898],{"className":11899,"style":443},[442],[394,11901,11903],{"className":11902},[447,448,449,450],[394,11904,520],{"className":11905},[415,450],[394,11907,983],{"className":11908},[982],[394,11910,11912],{"className":11911},[431],[394,11913,11915],{"className":11914,"style":5035},[435],[394,11916],{},[394,11918],{"className":11919},[465,524],[394,11921],{"className":11922,"style":5694},[465],[394,11924,11926,11929,11932,11941,11944,11984,11987,11990,11993,11996,11999,12002,12005,12008,12011,12014,12017,12020],{"className":11925},[406],[394,11927],{"className":11928,"style":535},[410],[394,11930,540],{"className":11931},[539],[394,11933,11935],{"className":11934},[415],[394,11936,11938],{"className":11937},[415],[394,11939,551],{"className":11940},[415,550],[394,11942],{"className":11943,"style":555},[465],[394,11945,11947,11950],{"className":11946},[415],[394,11948,3715],{"className":11949},[415,419],[394,11951,11953],{"className":11952},[423],[394,11954,11956,11976],{"className":11955},[427,913],[394,11957,11959,11973],{"className":11958},[431],[394,11960,11962],{"className":11961,"style":5013},[435],[394,11963,11964,11967],{"style":5016},[394,11965],{"className":11966,"style":443},[442],[394,11968,11970],{"className":11969},[447,448,449,450],[394,11971,520],{"className":11972},[415,450],[394,11974,983],{"className":11975},[982],[394,11977,11979],{"className":11978},[431],[394,11980,11982],{"className":11981,"style":5035},[435],[394,11983],{},[394,11985,563],{"className":11986},[562],[394,11988,769],{"className":11989},[768],[394,11991],{"className":11992,"style":5694},[465],[394,11994],{"className":11995,"style":734},[465],[394,11997,2418],{"className":11998},[738],[394,12000],{"className":12001,"style":734},[465],[394,12003,769],{"className":12004},[768],[394,12006],{"className":12007,"style":5694},[465],[394,12009],{"className":12010,"style":734},[465],[394,12012,4243],{"className":12013},[415,419],[394,12015],{"className":12016,"style":466},[465],[394,12018,471],{"className":12019},[470],[394,12021],{"className":12022,"style":466},[465],[394,12024,12026,12029,12069,12072],{"className":12025},[406],[394,12027],{"className":12028,"style":5126},[410],[394,12030,12032,12035],{"className":12031},[415],[394,12033,384],{"className":12034},[415,419],[394,12036,12038],{"className":12037},[423],[394,12039,12041,12061],{"className":12040},[427,913],[394,12042,12044,12058],{"className":12043},[431],[394,12045,12047],{"className":12046,"style":11532},[435],[394,12048,12049,12052],{"style":5016},[394,12050],{"className":12051,"style":443},[442],[394,12053,12055],{"className":12054},[447,448,449,450],[394,12056,605],{"className":12057},[415,419,450],[394,12059,983],{"className":12060},[982],[394,12062,12064],{"className":12063},[431],[394,12065,12067],{"className":12066,"style":5035},[435],[394,12068],{},[394,12070],{"className":12071},[465,524],[394,12073],{"className":12074,"style":5694},[465],[394,12076,12078,12081,12084,12093,12096,12136],{"className":12077},[406],[394,12079],{"className":12080,"style":535},[410],[394,12082,540],{"className":12083},[539],[394,12085,12087],{"className":12086},[415],[394,12088,12090],{"className":12089},[415],[394,12091,551],{"className":12092},[415,550],[394,12094],{"className":12095,"style":555},[465],[394,12097,12099,12102],{"className":12098},[415],[394,12100,3715],{"className":12101},[415,419],[394,12103,12105],{"className":12104},[423],[394,12106,12108,12128],{"className":12107},[427,913],[394,12109,12111,12125],{"className":12110},[431],[394,12112,12114],{"className":12113,"style":11532},[435],[394,12115,12116,12119],{"style":5016},[394,12117],{"className":12118,"style":443},[442],[394,12120,12122],{"className":12121},[447,448,449,450],[394,12123,605],{"className":12124},[415,419,450],[394,12126,983],{"className":12127},[982],[394,12129,12131],{"className":12130},[431],[394,12132,12134],{"className":12133,"style":5035},[435],[394,12135],{},[394,12137,563],{"className":12138},[562],[381,12140,12141,12142,12145,12146,12149,12150,1099,12265,12272,12273,12528,12529,12582,12583,12635,12636,12688,12689,12741,12742,12794,12795,12997,12998,13110,13111,13163],{},"with the moduli ",[389,12143,12144],{},"pairwise coprime",", CRT guarantees a ",[389,12147,12148],{},"unique"," solution modulo\n",[394,12151,12153],{"className":12152},[397],[394,12154,12156,12175],{"className":12155,"ariaHidden":402},[401],[394,12157,12159,12162,12166,12169,12172],{"className":12158},[406],[394,12160],{"className":12161,"style":871},[410],[394,12163,12165],{"className":12164,"style":7567},[415,419],"M",[394,12167],{"className":12168,"style":466},[465],[394,12170,674],{"className":12171},[470],[394,12173],{"className":12174,"style":466},[465],[394,12176,12178,12182,12222,12225],{"className":12177},[406],[394,12179],{"className":12180,"style":12181},[410],"height:1.0497em;vertical-align:-0.2997em;",[394,12183,12185,12188],{"className":12184},[4314],[394,12186,8794],{"className":12187,"style":4752},[4314,4359,4751],[394,12189,12191],{"className":12190},[423],[394,12192,12194,12214],{"className":12193},[427,913],[394,12195,12197,12211],{"className":12196},[431],[394,12198,12200],{"className":12199,"style":8807},[435],[394,12201,12202,12205],{"style":4768},[394,12203],{"className":12204,"style":443},[442],[394,12206,12208],{"className":12207},[447,448,449,450],[394,12209,5212],{"className":12210},[415,419,450],[394,12212,983],{"className":12213},[982],[394,12215,12217],{"className":12216},[431],[394,12218,12220],{"className":12219,"style":4787},[435],[394,12221],{},[394,12223],{"className":12224,"style":734},[465],[394,12226,12228,12231],{"className":12227},[415],[394,12229,3715],{"className":12230},[415,419],[394,12232,12234],{"className":12233},[423],[394,12235,12237,12257],{"className":12236},[427,913],[394,12238,12240,12254],{"className":12239},[431],[394,12241,12243],{"className":12242,"style":5200},[435],[394,12244,12245,12248],{"style":5016},[394,12246],{"className":12247,"style":443},[442],[394,12249,12251],{"className":12250},[447,448,449,450],[394,12252,5212],{"className":12253},[415,419,450],[394,12255,983],{"className":12256},[982],[394,12258,12260],{"className":12259},[431],[394,12261,12263],{"className":12262,"style":5035},[435],[394,12264],{},[1609,12266,12267],{},[384,12268,3330],{"href":12269,"ariaDescribedBy":12270,"dataFootnoteRef":376,"id":12271},"#user-content-fn-clrs-crt",[1615],"user-content-fnref-clrs-crt"," The construction is explicit and again uses the modular\ninverse. Let ",[394,12274,12276],{"className":12275},[397],[394,12277,12279,12335,12396],{"className":12278,"ariaHidden":402},[401],[394,12280,12282,12285,12326,12329,12332],{"className":12281},[406],[394,12283],{"className":12284,"style":11437},[410],[394,12286,12288,12291],{"className":12287},[415],[394,12289,12165],{"className":12290,"style":7567},[415,419],[394,12292,12294],{"className":12293},[423],[394,12295,12297,12318],{"className":12296},[427,913],[394,12298,12300,12315],{"className":12299},[431],[394,12301,12303],{"className":12302,"style":5200},[435],[394,12304,12306,12309],{"style":12305},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[394,12307],{"className":12308,"style":443},[442],[394,12310,12312],{"className":12311},[447,448,449,450],[394,12313,5212],{"className":12314},[415,419,450],[394,12316,983],{"className":12317},[982],[394,12319,12321],{"className":12320},[431],[394,12322,12324],{"className":12323,"style":5035},[435],[394,12325],{},[394,12327],{"className":12328,"style":466},[465],[394,12330,674],{"className":12331},[470],[394,12333],{"className":12334,"style":466},[465],[394,12336,12338,12341,12344,12347,12387,12390,12393],{"className":12337},[406],[394,12339],{"className":12340,"style":535},[410],[394,12342,12165],{"className":12343,"style":7567},[415,419],[394,12345,11685],{"className":12346},[415],[394,12348,12350,12353],{"className":12349},[415],[394,12351,3715],{"className":12352},[415,419],[394,12354,12356],{"className":12355},[423],[394,12357,12359,12379],{"className":12358},[427,913],[394,12360,12362,12376],{"className":12361},[431],[394,12363,12365],{"className":12364,"style":5200},[435],[394,12366,12367,12370],{"style":5016},[394,12368],{"className":12369,"style":443},[442],[394,12371,12373],{"className":12372},[447,448,449,450],[394,12374,5212],{"className":12375},[415,419,450],[394,12377,983],{"className":12378},[982],[394,12380,12382],{"className":12381},[431],[394,12383,12385],{"className":12384,"style":5035},[435],[394,12386],{},[394,12388],{"className":12389,"style":466},[465],[394,12391,674],{"className":12392},[470],[394,12394],{"className":12395,"style":466},[465],[394,12397,12399,12403,12485,12488],{"className":12398},[406],[394,12400],{"className":12401,"style":12402},[410],"height:1.1858em;vertical-align:-0.4358em;",[394,12404,12406,12409],{"className":12405},[4314],[394,12407,8794],{"className":12408,"style":4752},[4314,4359,4751],[394,12410,12412],{"className":12411},[423],[394,12413,12415,12477],{"className":12414},[427,913],[394,12416,12418,12474],{"className":12417},[431],[394,12419,12421],{"className":12420,"style":4765},[435],[394,12422,12423,12426],{"style":4768},[394,12424],{"className":12425,"style":443},[442],[394,12427,12429],{"className":12428},[447,448,449,450],[394,12430,12432,12435,12471],{"className":12431},[415,450],[394,12433,9432],{"className":12434,"style":9431},[415,419,450],[394,12436,12438,12465,12468],{"className":12437},[470,450],[394,12439,12441],{"className":12440},[470,450],[394,12442,12444],{"className":12443},[415,7399,450],[394,12445,12447],{"className":12446},[7403,450],[394,12448,12450,12453,12462],{"className":12449},[7407,450],[394,12451],{"className":12452,"style":761},[410],[394,12454,12456],{"className":12455},[7414],[394,12457,12459],{"className":12458},[415,450],[394,12460,7421],{"className":12461},[470,450],[394,12463],{"className":12464},[7425],[394,12466],{"className":12467},[465,7429,450],[394,12469,674],{"className":12470},[470,450],[394,12472,5212],{"className":12473},[415,419,450],[394,12475,983],{"className":12476},[982],[394,12478,12480],{"className":12479},[431],[394,12481,12483],{"className":12482,"style":10163},[435],[394,12484],{},[394,12486],{"className":12487,"style":734},[465],[394,12489,12491,12494],{"className":12490},[415],[394,12492,3715],{"className":12493},[415,419],[394,12495,12497],{"className":12496},[423],[394,12498,12500,12520],{"className":12499},[427,913],[394,12501,12503,12517],{"className":12502},[431],[394,12504,12506],{"className":12505,"style":5200},[435],[394,12507,12508,12511],{"style":5016},[394,12509],{"className":12510,"style":443},[442],[394,12512,12514],{"className":12513},[447,448,449,450],[394,12515,9432],{"className":12516,"style":9431},[415,419,450],[394,12518,983],{"className":12519},[982],[394,12521,12523],{"className":12522},[431],[394,12524,12526],{"className":12525,"style":9556},[435],[394,12527],{},". Because the ",[394,12530,12532],{"className":12531},[397],[394,12533,12535],{"className":12534,"ariaHidden":402},[401],[394,12536,12538,12542],{"className":12537},[406],[394,12539],{"className":12540,"style":12541},[410],"height:0.7167em;vertical-align:-0.2861em;",[394,12543,12545,12548],{"className":12544},[415],[394,12546,3715],{"className":12547},[415,419],[394,12549,12551],{"className":12550},[423],[394,12552,12554,12574],{"className":12553},[427,913],[394,12555,12557,12571],{"className":12556},[431],[394,12558,12560],{"className":12559,"style":5200},[435],[394,12561,12562,12565],{"style":5016},[394,12563],{"className":12564,"style":443},[442],[394,12566,12568],{"className":12567},[447,448,449,450],[394,12569,9432],{"className":12570,"style":9431},[415,419,450],[394,12572,983],{"className":12573},[982],[394,12575,12577],{"className":12576},[431],[394,12578,12580],{"className":12579,"style":9556},[435],[394,12581],{}," are coprime to\n",[394,12584,12586],{"className":12585},[397],[394,12587,12589],{"className":12588,"ariaHidden":402},[401],[394,12590,12592,12595],{"className":12591},[406],[394,12593],{"className":12594,"style":5126},[410],[394,12596,12598,12601],{"className":12597},[415],[394,12599,3715],{"className":12600},[415,419],[394,12602,12604],{"className":12603},[423],[394,12605,12607,12627],{"className":12606},[427,913],[394,12608,12610,12624],{"className":12609},[431],[394,12611,12613],{"className":12612,"style":5200},[435],[394,12614,12615,12618],{"style":5016},[394,12616],{"className":12617,"style":443},[442],[394,12619,12621],{"className":12620},[447,448,449,450],[394,12622,5212],{"className":12623},[415,419,450],[394,12625,983],{"className":12626},[982],[394,12628,12630],{"className":12629},[431],[394,12631,12633],{"className":12632,"style":5035},[435],[394,12634],{},", so is ",[394,12637,12639],{"className":12638},[397],[394,12640,12642],{"className":12641,"ariaHidden":402},[401],[394,12643,12645,12648],{"className":12644},[406],[394,12646],{"className":12647,"style":11437},[410],[394,12649,12651,12654],{"className":12650},[415],[394,12652,12165],{"className":12653,"style":7567},[415,419],[394,12655,12657],{"className":12656},[423],[394,12658,12660,12680],{"className":12659},[427,913],[394,12661,12663,12677],{"className":12662},[431],[394,12664,12666],{"className":12665,"style":5200},[435],[394,12667,12668,12671],{"style":12305},[394,12669],{"className":12670,"style":443},[442],[394,12672,12674],{"className":12673},[447,448,449,450],[394,12675,5212],{"className":12676},[415,419,450],[394,12678,983],{"className":12679},[982],[394,12681,12683],{"className":12682},[431],[394,12684,12686],{"className":12685,"style":5035},[435],[394,12687],{},", hence ",[394,12690,12692],{"className":12691},[397],[394,12693,12695],{"className":12694,"ariaHidden":402},[401],[394,12696,12698,12701],{"className":12697},[406],[394,12699],{"className":12700,"style":11437},[410],[394,12702,12704,12707],{"className":12703},[415],[394,12705,12165],{"className":12706,"style":7567},[415,419],[394,12708,12710],{"className":12709},[423],[394,12711,12713,12733],{"className":12712},[427,913],[394,12714,12716,12730],{"className":12715},[431],[394,12717,12719],{"className":12718,"style":5200},[435],[394,12720,12721,12724],{"style":12305},[394,12722],{"className":12723,"style":443},[442],[394,12725,12727],{"className":12726},[447,448,449,450],[394,12728,5212],{"className":12729},[415,419,450],[394,12731,983],{"className":12732},[982],[394,12734,12736],{"className":12735},[431],[394,12737,12739],{"className":12738,"style":5035},[435],[394,12740],{}," has an inverse modulo ",[394,12743,12745],{"className":12744},[397],[394,12746,12748],{"className":12747,"ariaHidden":402},[401],[394,12749,12751,12754],{"className":12750},[406],[394,12752],{"className":12753,"style":5126},[410],[394,12755,12757,12760],{"className":12756},[415],[394,12758,3715],{"className":12759},[415,419],[394,12761,12763],{"className":12762},[423],[394,12764,12766,12786],{"className":12765},[427,913],[394,12767,12769,12783],{"className":12768},[431],[394,12770,12772],{"className":12771,"style":5200},[435],[394,12773,12774,12777],{"style":5016},[394,12775],{"className":12776,"style":443},[442],[394,12778,12780],{"className":12779},[447,448,449,450],[394,12781,5212],{"className":12782},[415,419,450],[394,12784,983],{"className":12785},[982],[394,12787,12789],{"className":12788},[431],[394,12790,12792],{"className":12791,"style":5035},[435],[394,12793],{},"; call it\n",[394,12796,12798],{"className":12797},[397],[394,12799,12801,12878],{"className":12800,"ariaHidden":402},[401],[394,12802,12804,12808,12869,12872,12875],{"className":12803},[406],[394,12805],{"className":12806,"style":12807},[410],"height:1.1311em;vertical-align:-0.2769em;",[394,12809,12811,12814],{"className":12810},[415],[394,12812,12165],{"className":12813,"style":7567},[415,419],[394,12815,12817],{"className":12816},[423],[394,12818,12820,12860],{"className":12819},[427,913],[394,12821,12823,12857],{"className":12822},[431],[394,12824,12827,12839],{"className":12825,"style":12826},[435],"height:0.8542em;",[394,12828,12830,12833],{"style":12829},"top:-2.4231em;margin-left:-0.109em;margin-right:0.05em;",[394,12831],{"className":12832,"style":443},[442],[394,12834,12836],{"className":12835},[447,448,449,450],[394,12837,5212],{"className":12838},[415,419,450],[394,12840,12842,12845],{"style":12841},"top:-3.1031em;margin-right:0.05em;",[394,12843],{"className":12844,"style":443},[442],[394,12846,12848],{"className":12847},[447,448,449,450],[394,12849,12851,12854],{"className":12850},[415,450],[394,12852,457],{"className":12853},[415,450],[394,12855,461],{"className":12856},[415,450],[394,12858,983],{"className":12859},[982],[394,12861,12863],{"className":12862},[431],[394,12864,12867],{"className":12865,"style":12866},[435],"height:0.2769em;",[394,12868],{},[394,12870],{"className":12871,"style":466},[465],[394,12873,471],{"className":12874},[470],[394,12876],{"className":12877,"style":466},[465],[394,12879,12881,12885],{"className":12880},[406],[394,12882],{"className":12883,"style":12884},[410],"height:1.3217em;vertical-align:-0.2769em;",[394,12886,12888,12891],{"className":12887},[415],[394,12889,12165],{"className":12890,"style":7567},[415,419],[394,12892,12894],{"className":12893},[423],[394,12895,12897,12989],{"className":12896},[427,913],[394,12898,12900,12986],{"className":12899},[431],[394,12901,12904,12915],{"className":12902,"style":12903},[435],"height:1.0448em;",[394,12905,12906,12909],{"style":12829},[394,12907],{"className":12908,"style":443},[442],[394,12910,12912],{"className":12911},[447,448,449,450],[394,12913,5212],{"className":12914},[415,419,450],[394,12916,12918,12921],{"style":12917},"top:-3.2198em;margin-right:0.05em;",[394,12919],{"className":12920,"style":443},[442],[394,12922,12924],{"className":12923},[447,448,449,450],[394,12925,12927,12930,12934,12937,12977,12980,12983],{"className":12926},[415,450],[394,12928],{"className":12929,"style":3711},[465,450],[394,12931,12933],{"className":12932},[415,419,450],"φ",[394,12935,540],{"className":12936},[539,450],[394,12938,12940,12943],{"className":12939},[415,450],[394,12941,3715],{"className":12942},[415,419,450],[394,12944,12946],{"className":12945},[423],[394,12947,12949,12969],{"className":12948},[427,913],[394,12950,12952,12966],{"className":12951},[431],[394,12953,12955],{"className":12954,"style":8883},[435],[394,12956,12957,12960],{"style":8941},[394,12958],{"className":12959,"style":8890},[442],[394,12961,12963],{"className":12962},[447,8894,1507,450],[394,12964,5212],{"className":12965},[415,419,450],[394,12967,983],{"className":12968},[982],[394,12970,12972],{"className":12971},[431],[394,12973,12975],{"className":12974,"style":8907},[435],[394,12976],{},[394,12978,563],{"className":12979},[562,450],[394,12981,457],{"className":12982},[516,450],[394,12984,461],{"className":12985},[415,450],[394,12987,983],{"className":12988},[982],[394,12990,12992],{"className":12991},[431],[394,12993,12995],{"className":12994,"style":12866},[435],[394,12996],{}," (or ",[394,12999,13001],{"className":13000},[397],[394,13002,13004],{"className":13003,"ariaHidden":402},[401],[394,13005,13007,13011],{"className":13006},[406],[394,13008],{"className":13009,"style":13010},[410],"height:1.1729em;vertical-align:-0.2769em;",[394,13012,13014,13017],{"className":13013},[415],[394,13015,12165],{"className":13016,"style":7567},[415,419],[394,13018,13020],{"className":13019},[423],[394,13021,13023,13102],{"className":13022},[427,913],[394,13024,13026,13099],{"className":13025},[431],[394,13027,13030,13041],{"className":13028,"style":13029},[435],"height:0.896em;",[394,13031,13032,13035],{"style":12829},[394,13033],{"className":13034,"style":443},[442],[394,13036,13038],{"className":13037},[447,448,449,450],[394,13039,5212],{"className":13040},[415,419,450],[394,13042,13044,13047],{"style":13043},"top:-3.1449em;margin-right:0.05em;",[394,13045],{"className":13046,"style":443},[442],[394,13048,13050],{"className":13049},[447,448,449,450],[394,13051,13053,13093,13096],{"className":13052},[415,450],[394,13054,13056,13059],{"className":13055},[415,450],[394,13057,3715],{"className":13058},[415,419,450],[394,13060,13062],{"className":13061},[423],[394,13063,13065,13085],{"className":13064},[427,913],[394,13066,13068,13082],{"className":13067},[431],[394,13069,13071],{"className":13070,"style":8883},[435],[394,13072,13073,13076],{"style":8941},[394,13074],{"className":13075,"style":8890},[442],[394,13077,13079],{"className":13078},[447,8894,1507,450],[394,13080,5212],{"className":13081},[415,419,450],[394,13083,983],{"className":13084},[982],[394,13086,13088],{"className":13087},[431],[394,13089,13091],{"className":13090,"style":8907},[435],[394,13092],{},[394,13094,457],{"className":13095},[516,450],[394,13097,520],{"className":13098},[415,450],[394,13100,983],{"className":13101},[982],[394,13103,13105],{"className":13104},[431],[394,13106,13108],{"className":13107,"style":12866},[435],[394,13109],{}," when ",[394,13112,13114],{"className":13113},[397],[394,13115,13117],{"className":13116,"ariaHidden":402},[401],[394,13118,13120,13123],{"className":13119},[406],[394,13121],{"className":13122,"style":5126},[410],[394,13124,13126,13129],{"className":13125},[415],[394,13127,3715],{"className":13128},[415,419],[394,13130,13132],{"className":13131},[423],[394,13133,13135,13155],{"className":13134},[427,913],[394,13136,13138,13152],{"className":13137},[431],[394,13139,13141],{"className":13140,"style":5200},[435],[394,13142,13143,13146],{"style":5016},[394,13144],{"className":13145,"style":443},[442],[394,13147,13149],{"className":13148},[447,448,449,450],[394,13150,5212],{"className":13151},[415,419,450],[394,13153,983],{"className":13154},[982],[394,13156,13158],{"className":13157},[431],[394,13159,13161],{"className":13160,"style":5035},[435],[394,13162],{}," is prime). Then",[394,13165,13167],{"className":13166},[647],[394,13168,13170],{"className":13169},[397],[394,13171,13173,13191,13418],{"className":13172,"ariaHidden":402},[401],[394,13174,13176,13179,13182,13185,13188],{"className":13175},[406],[394,13177],{"className":13178,"style":11721},[410],[394,13180,4243],{"className":13181},[415,419],[394,13183],{"className":13184,"style":466},[465],[394,13186,471],{"className":13187},[470],[394,13189],{"className":13190,"style":466},[465],[394,13192,13194,13197,13264,13267,13307,13310,13350,13353,13412,13415],{"className":13193},[406],[394,13195],{"className":13196,"style":9062},[410],[394,13198,13200],{"className":13199},[4314,4315],[394,13201,13203,13256],{"className":13202},[427,913],[394,13204,13206,13253],{"className":13205},[431],[394,13207,13209,13229,13239],{"className":13208,"style":4325},[435],[394,13210,13211,13214],{"style":9127},[394,13212],{"className":13213,"style":4332},[442],[394,13215,13217],{"className":13216},[447,448,449,450],[394,13218,13220,13223,13226],{"className":13219},[415,450],[394,13221,5212],{"className":13222},[415,419,450],[394,13224,674],{"className":13225},[470,450],[394,13227,461],{"className":13228},[415,450],[394,13230,13231,13234],{"style":4350},[394,13232],{"className":13233,"style":4332},[442],[394,13235,13236],{},[394,13237,4361],{"className":13238},[4314,4359,4360],[394,13240,13241,13244],{"style":4364},[394,13242],{"className":13243,"style":4332},[442],[394,13245,13247],{"className":13246},[447,448,449,450],[394,13248,13250],{"className":13249},[415,450],[394,13251,605],{"className":13252},[415,419,450],[394,13254,983],{"className":13255},[982],[394,13257,13259],{"className":13258},[431],[394,13260,13262],{"className":13261,"style":9180},[435],[394,13263],{},[394,13265],{"className":13266,"style":734},[465],[394,13268,13270,13273],{"className":13269},[415],[394,13271,384],{"className":13272},[415,419],[394,13274,13276],{"className":13275},[423],[394,13277,13279,13299],{"className":13278},[427,913],[394,13280,13282,13296],{"className":13281},[431],[394,13283,13285],{"className":13284,"style":5200},[435],[394,13286,13287,13290],{"style":5016},[394,13288],{"className":13289,"style":443},[442],[394,13291,13293],{"className":13292},[447,448,449,450],[394,13294,5212],{"className":13295},[415,419,450],[394,13297,983],{"className":13298},[982],[394,13300,13302],{"className":13301},[431],[394,13303,13305],{"className":13304,"style":5035},[435],[394,13306],{},[394,13308],{"className":13309,"style":734},[465],[394,13311,13313,13316],{"className":13312},[415],[394,13314,12165],{"className":13315,"style":7567},[415,419],[394,13317,13319],{"className":13318},[423],[394,13320,13322,13342],{"className":13321},[427,913],[394,13323,13325,13339],{"className":13324},[431],[394,13326,13328],{"className":13327,"style":5200},[435],[394,13329,13330,13333],{"style":12305},[394,13331],{"className":13332,"style":443},[442],[394,13334,13336],{"className":13335},[447,448,449,450],[394,13337,5212],{"className":13338},[415,419,450],[394,13340,983],{"className":13341},[982],[394,13343,13345],{"className":13344},[431],[394,13346,13348],{"className":13347,"style":5035},[435],[394,13349],{},[394,13351],{"className":13352,"style":734},[465],[394,13354,13356,13359],{"className":13355},[415],[394,13357,12165],{"className":13358,"style":7567},[415,419],[394,13360,13362],{"className":13361},[423],[394,13363,13365,13403],{"className":13364},[427,913],[394,13366,13368,13400],{"className":13367},[431],[394,13369,13371,13383],{"className":13370,"style":9877},[435],[394,13372,13374,13377],{"style":13373},"top:-2.433em;margin-left:-0.109em;margin-right:0.05em;",[394,13375],{"className":13376,"style":443},[442],[394,13378,13380],{"className":13379},[447,448,449,450],[394,13381,5212],{"className":13382},[415,419,450],[394,13384,13385,13388],{"style":4285},[394,13386],{"className":13387,"style":443},[442],[394,13389,13391],{"className":13390},[447,448,449,450],[394,13392,13394,13397],{"className":13393},[415,450],[394,13395,457],{"className":13396},[415,450],[394,13398,461],{"className":13399},[415,450],[394,13401,983],{"className":13402},[982],[394,13404,13406],{"className":13405},[431],[394,13407,13410],{"className":13408,"style":13409},[435],"height:0.267em;",[394,13411],{},[394,13413],{"className":13414},[465,524],[394,13416],{"className":13417,"style":5694},[465],[394,13419,13421,13424,13427,13436,13439,13442,13445],{"className":13420},[406],[394,13422],{"className":13423,"style":535},[410],[394,13425,540],{"className":13426},[539],[394,13428,13430],{"className":13429},[415],[394,13431,13433],{"className":13432},[415],[394,13434,551],{"className":13435},[415,550],[394,13437],{"className":13438,"style":555},[465],[394,13440,12165],{"className":13441,"style":7567},[415,419],[394,13443,563],{"className":13444},[562],[394,13446,1099],{"className":13447},[415],[381,13449,13450,13451,13600,13601,13735,13736,13863,13864,13891,13892,13735,13944,14051,14052,14067,14068,6249,14113,14158,14159,14162,14163,14229],{},"Each term ",[394,13452,13454],{"className":13453},[397],[394,13455,13457],{"className":13456,"ariaHidden":402},[401],[394,13458,13460,13463,13503,13543],{"className":13459},[406],[394,13461],{"className":13462,"style":12807},[410],[394,13464,13466,13469],{"className":13465},[415],[394,13467,384],{"className":13468},[415,419],[394,13470,13472],{"className":13471},[423],[394,13473,13475,13495],{"className":13474},[427,913],[394,13476,13478,13492],{"className":13477},[431],[394,13479,13481],{"className":13480,"style":5200},[435],[394,13482,13483,13486],{"style":5016},[394,13484],{"className":13485,"style":443},[442],[394,13487,13489],{"className":13488},[447,448,449,450],[394,13490,5212],{"className":13491},[415,419,450],[394,13493,983],{"className":13494},[982],[394,13496,13498],{"className":13497},[431],[394,13499,13501],{"className":13500,"style":5035},[435],[394,13502],{},[394,13504,13506,13509],{"className":13505},[415],[394,13507,12165],{"className":13508,"style":7567},[415,419],[394,13510,13512],{"className":13511},[423],[394,13513,13515,13535],{"className":13514},[427,913],[394,13516,13518,13532],{"className":13517},[431],[394,13519,13521],{"className":13520,"style":5200},[435],[394,13522,13523,13526],{"style":12305},[394,13524],{"className":13525,"style":443},[442],[394,13527,13529],{"className":13528},[447,448,449,450],[394,13530,5212],{"className":13531},[415,419,450],[394,13533,983],{"className":13534},[982],[394,13536,13538],{"className":13537},[431],[394,13539,13541],{"className":13540,"style":5035},[435],[394,13542],{},[394,13544,13546,13549],{"className":13545},[415],[394,13547,12165],{"className":13548,"style":7567},[415,419],[394,13550,13552],{"className":13551},[423],[394,13553,13555,13592],{"className":13554},[427,913],[394,13556,13558,13589],{"className":13557},[431],[394,13559,13561,13572],{"className":13560,"style":12826},[435],[394,13562,13563,13566],{"style":12829},[394,13564],{"className":13565,"style":443},[442],[394,13567,13569],{"className":13568},[447,448,449,450],[394,13570,5212],{"className":13571},[415,419,450],[394,13573,13574,13577],{"style":12841},[394,13575],{"className":13576,"style":443},[442],[394,13578,13580],{"className":13579},[447,448,449,450],[394,13581,13583,13586],{"className":13582},[415,450],[394,13584,457],{"className":13585},[415,450],[394,13587,461],{"className":13588},[415,450],[394,13590,983],{"className":13591},[982],[394,13593,13595],{"className":13594},[431],[394,13596,13598],{"className":13597,"style":12866},[435],[394,13599],{}," is ",[394,13602,13604],{"className":13603},[397],[394,13605,13607,13619,13671],{"className":13606,"ariaHidden":402},[401],[394,13608,13610,13613,13616],{"className":13609},[406],[394,13611],{"className":13612,"style":11721},[410],[394,13614,471],{"className":13615},[470],[394,13617],{"className":13618,"style":466},[465],[394,13620,13622,13625,13665,13668],{"className":13621},[406],[394,13623],{"className":13624,"style":5126},[410],[394,13626,13628,13631],{"className":13627},[415],[394,13629,384],{"className":13630},[415,419],[394,13632,13634],{"className":13633},[423],[394,13635,13637,13657],{"className":13636},[427,913],[394,13638,13640,13654],{"className":13639},[431],[394,13641,13643],{"className":13642,"style":5200},[435],[394,13644,13645,13648],{"style":5016},[394,13646],{"className":13647,"style":443},[442],[394,13649,13651],{"className":13650},[447,448,449,450],[394,13652,5212],{"className":13653},[415,419,450],[394,13655,983],{"className":13656},[982],[394,13658,13660],{"className":13659},[431],[394,13661,13663],{"className":13662,"style":5035},[435],[394,13664],{},[394,13666],{"className":13667},[465,524],[394,13669],{"className":13670,"style":528},[465],[394,13672,13674,13677,13680,13689,13692,13732],{"className":13673},[406],[394,13675],{"className":13676,"style":535},[410],[394,13678,540],{"className":13679},[539],[394,13681,13683],{"className":13682},[415],[394,13684,13686],{"className":13685},[415],[394,13687,551],{"className":13688},[415,550],[394,13690],{"className":13691,"style":555},[465],[394,13693,13695,13698],{"className":13694},[415],[394,13696,3715],{"className":13697},[415,419],[394,13699,13701],{"className":13700},[423],[394,13702,13704,13724],{"className":13703},[427,913],[394,13705,13707,13721],{"className":13706},[431],[394,13708,13710],{"className":13709,"style":5200},[435],[394,13711,13712,13715],{"style":5016},[394,13713],{"className":13714,"style":443},[442],[394,13716,13718],{"className":13717},[447,448,449,450],[394,13719,5212],{"className":13720},[415,419,450],[394,13722,983],{"className":13723},[982],[394,13725,13727],{"className":13726},[431],[394,13728,13730],{"className":13729,"style":5035},[435],[394,13731],{},[394,13733,563],{"className":13734},[562]," (since ",[394,13737,13739],{"className":13738},[397],[394,13740,13742,13854],{"className":13741,"ariaHidden":402},[401],[394,13743,13745,13748,13788,13845,13848,13851],{"className":13744},[406],[394,13746],{"className":13747,"style":12807},[410],[394,13749,13751,13754],{"className":13750},[415],[394,13752,12165],{"className":13753,"style":7567},[415,419],[394,13755,13757],{"className":13756},[423],[394,13758,13760,13780],{"className":13759},[427,913],[394,13761,13763,13777],{"className":13762},[431],[394,13764,13766],{"className":13765,"style":5200},[435],[394,13767,13768,13771],{"style":12305},[394,13769],{"className":13770,"style":443},[442],[394,13772,13774],{"className":13773},[447,448,449,450],[394,13775,5212],{"className":13776},[415,419,450],[394,13778,983],{"className":13779},[982],[394,13781,13783],{"className":13782},[431],[394,13784,13786],{"className":13785,"style":5035},[435],[394,13787],{},[394,13789,13791,13794],{"className":13790},[415],[394,13792,12165],{"className":13793,"style":7567},[415,419],[394,13795,13797],{"className":13796},[423],[394,13798,13800,13837],{"className":13799},[427,913],[394,13801,13803,13834],{"className":13802},[431],[394,13804,13806,13817],{"className":13805,"style":12826},[435],[394,13807,13808,13811],{"style":12829},[394,13809],{"className":13810,"style":443},[442],[394,13812,13814],{"className":13813},[447,448,449,450],[394,13815,5212],{"className":13816},[415,419,450],[394,13818,13819,13822],{"style":12841},[394,13820],{"className":13821,"style":443},[442],[394,13823,13825],{"className":13824},[447,448,449,450],[394,13826,13828,13831],{"className":13827},[415,450],[394,13829,457],{"className":13830},[415,450],[394,13832,461],{"className":13833},[415,450],[394,13835,983],{"className":13836},[982],[394,13838,13840],{"className":13839},[431],[394,13841,13843],{"className":13842,"style":12866},[435],[394,13844],{},[394,13846],{"className":13847,"style":466},[465],[394,13849,471],{"className":13850},[470],[394,13852],{"className":13853,"style":466},[465],[394,13855,13857,13860],{"className":13856},[406],[394,13858],{"className":13859,"style":639},[410],[394,13861,461],{"className":13862},[415],"\nthere) and ",[394,13865,13867],{"className":13866},[397],[394,13868,13870,13882],{"className":13869,"ariaHidden":402},[401],[394,13871,13873,13876,13879],{"className":13872},[406],[394,13874],{"className":13875,"style":11721},[410],[394,13877,471],{"className":13878},[470],[394,13880],{"className":13881,"style":466},[465],[394,13883,13885,13888],{"className":13884},[406],[394,13886],{"className":13887,"style":639},[410],[394,13889,780],{"className":13890},[415]," modulo every other ",[394,13893,13895],{"className":13894},[397],[394,13896,13898],{"className":13897,"ariaHidden":402},[401],[394,13899,13901,13904],{"className":13900},[406],[394,13902],{"className":13903,"style":12541},[410],[394,13905,13907,13910],{"className":13906},[415],[394,13908,3715],{"className":13909},[415,419],[394,13911,13913],{"className":13912},[423],[394,13914,13916,13936],{"className":13915},[427,913],[394,13917,13919,13933],{"className":13918},[431],[394,13920,13922],{"className":13921,"style":5200},[435],[394,13923,13924,13927],{"style":5016},[394,13925],{"className":13926,"style":443},[442],[394,13928,13930],{"className":13929},[447,448,449,450],[394,13931,9432],{"className":13932,"style":9431},[415,419,450],[394,13934,983],{"className":13935},[982],[394,13937,13939],{"className":13938},[431],[394,13940,13942],{"className":13941,"style":9556},[435],[394,13943],{},[394,13945,13947],{"className":13946},[397],[394,13948,13950,14005],{"className":13949,"ariaHidden":402},[401],[394,13951,13953,13956,13996,13999,14002],{"className":13952},[406],[394,13954],{"className":13955,"style":9517},[410],[394,13957,13959,13962],{"className":13958},[415],[394,13960,3715],{"className":13961},[415,419],[394,13963,13965],{"className":13964},[423],[394,13966,13968,13988],{"className":13967},[427,913],[394,13969,13971,13985],{"className":13970},[431],[394,13972,13974],{"className":13973,"style":5200},[435],[394,13975,13976,13979],{"style":5016},[394,13977],{"className":13978,"style":443},[442],[394,13980,13982],{"className":13981},[447,448,449,450],[394,13983,9432],{"className":13984,"style":9431},[415,419,450],[394,13986,983],{"className":13987},[982],[394,13989,13991],{"className":13990},[431],[394,13992,13994],{"className":13993,"style":9556},[435],[394,13995],{},[394,13997],{"className":13998,"style":466},[465],[394,14000,5640],{"className":14001},[470],[394,14003],{"className":14004,"style":466},[465],[394,14006,14008,14011],{"className":14007},[406],[394,14009],{"className":14010,"style":11437},[410],[394,14012,14014,14017],{"className":14013},[415],[394,14015,12165],{"className":14016,"style":7567},[415,419],[394,14018,14020],{"className":14019},[423],[394,14021,14023,14043],{"className":14022},[427,913],[394,14024,14026,14040],{"className":14025},[431],[394,14027,14029],{"className":14028,"style":5200},[435],[394,14030,14031,14034],{"style":12305},[394,14032],{"className":14033,"style":443},[442],[394,14035,14037],{"className":14036},[447,448,449,450],[394,14038,5212],{"className":14039},[415,419,450],[394,14041,983],{"className":14042},[982],[394,14044,14046],{"className":14045},[431],[394,14047,14049],{"className":14048,"style":5035},[435],[394,14050],{},"), so the sum\nsatisfies all ",[394,14053,14055],{"className":14054},[397],[394,14056,14058],{"className":14057,"ariaHidden":402},[401],[394,14059,14061,14064],{"className":14060},[406],[394,14062],{"className":14063,"style":601},[410],[394,14065,605],{"className":14066},[415,419]," congruences simultaneously. Concretely, to solve\n",[394,14069,14071],{"className":14070},[397],[394,14072,14074,14092],{"className":14073,"ariaHidden":402},[401],[394,14075,14077,14080,14083,14086,14089],{"className":14076},[406],[394,14078],{"className":14079,"style":11721},[410],[394,14081,4243],{"className":14082},[415,419],[394,14084],{"className":14085,"style":466},[465],[394,14087,471],{"className":14088},[470],[394,14090],{"className":14091,"style":466},[465],[394,14093,14095,14098,14101,14104,14107,14110],{"className":14094},[406],[394,14096],{"className":14097,"style":535},[410],[394,14099,520],{"className":14100},[415],[394,14102],{"className":14103,"style":734},[465],[394,14105,540],{"className":14106},[539],[394,14108,3386],{"className":14109},[415],[394,14111,563],{"className":14112},[562],[394,14114,14116],{"className":14115},[397],[394,14117,14119,14137],{"className":14118,"ariaHidden":402},[401],[394,14120,14122,14125,14128,14131,14134],{"className":14121},[406],[394,14123],{"className":14124,"style":11721},[410],[394,14126,4243],{"className":14127},[415,419],[394,14129],{"className":14130,"style":466},[465],[394,14132,471],{"className":14133},[470],[394,14135],{"className":14136,"style":466},[465],[394,14138,14140,14143,14146,14149,14152,14155],{"className":14139},[406],[394,14141],{"className":14142,"style":535},[410],[394,14144,3386],{"className":14145},[415],[394,14147],{"className":14148,"style":734},[465],[394,14150,540],{"className":14151},[539],[394,14153,5573],{"className":14154},[415],[394,14156,563],{"className":14157},[562]," each term acts as a ",[389,14160,14161],{},"selector",": one lands\non its own residue and vanishes modulo the other, so adding them assembles the\nanswer ",[394,14164,14166],{"className":14165},[397],[394,14167,14169,14187,14202],{"className":14168,"ariaHidden":402},[401],[394,14170,14172,14175,14178,14181,14184],{"className":14171},[406],[394,14173],{"className":14174,"style":11721},[410],[394,14176,4243],{"className":14177},[415,419],[394,14179],{"className":14180,"style":466},[465],[394,14182,471],{"className":14183},[470],[394,14185],{"className":14186,"style":466},[465],[394,14188,14190,14193,14196,14199],{"className":14189},[406],[394,14191],{"className":14192,"style":639},[410],[394,14194,4024],{"className":14195},[415],[394,14197],{"className":14198},[465,524],[394,14200],{"className":14201,"style":528},[465],[394,14203,14205,14208,14211,14220,14223,14226],{"className":14204},[406],[394,14206],{"className":14207,"style":535},[410],[394,14209,540],{"className":14210},[539],[394,14212,14214],{"className":14213},[415],[394,14215,14217],{"className":14216},[415],[394,14218,551],{"className":14219},[415,550],[394,14221],{"className":14222,"style":555},[465],[394,14224,10660],{"className":14225},[415],[394,14227,563],{"className":14228},[562]," one congruence at a time.",[2817,14231,14233,14609],{"className":14232},[2820,2821],[2823,14234,14238],{"xmlns":2825,"width":14235,"height":14236,"viewBox":14237},"305.210","186.304","-75 -75 228.908 139.728",[2830,14239,14240,14261,14280,14283,14346,14354,14364,14421,14429,14438,14441,14480,14491,14502],{"stroke":2832,"style":2833},[2830,14241,14242,14249,14255],{"stroke":2837,"fontFamily":4023,"fontSize":4024},[2830,14243,14245],{"transform":14244},"translate(29.09 -17.14)",[2835,14246],{"d":14247,"fill":2832,"stroke":2832,"className":14248,"style":4032},"M39.467-38.127L37.611-38.127L37.611-38.424Q37.885-38.424 38.053-38.471Q38.221-38.518 38.221-38.686L38.221-40.822Q38.221-41.037 38.158-41.133Q38.096-41.229 37.977-41.250Q37.858-41.272 37.611-41.272L37.611-41.568L38.803-41.654L38.803-40.920Q38.916-41.135 39.110-41.303Q39.303-41.471 39.541-41.563Q39.779-41.654 40.033-41.654Q40.994-41.654 41.170-40.943Q41.354-41.272 41.682-41.463Q42.010-41.654 42.389-41.654Q43.565-41.654 43.565-40.576L43.565-38.686Q43.565-38.518 43.733-38.471Q43.901-38.424 44.170-38.424L44.170-38.127L42.315-38.127L42.315-38.424Q42.588-38.424 42.756-38.469Q42.924-38.514 42.924-38.686L42.924-40.561Q42.924-40.947 42.799-41.174Q42.674-41.400 42.322-41.400Q42.018-41.400 41.762-41.238Q41.506-41.076 41.358-40.807Q41.209-40.537 41.209-40.240L41.209-38.686Q41.209-38.518 41.379-38.471Q41.549-38.424 41.819-38.424L41.819-38.127L39.963-38.127L39.963-38.424Q40.236-38.424 40.404-38.471Q40.572-38.518 40.572-38.686L40.572-40.561Q40.572-40.947 40.447-41.174Q40.322-41.400 39.971-41.400Q39.666-41.400 39.410-41.238Q39.154-41.076 39.006-40.807Q38.858-40.537 38.858-40.240L38.858-38.686Q38.858-38.518 39.028-38.471Q39.197-38.424 39.467-38.424L39.467-38.127M44.615-39.822Q44.615-40.326 44.871-40.758Q45.127-41.190 45.563-41.441Q45.998-41.693 46.498-41.693Q46.885-41.693 47.227-41.549Q47.569-41.404 47.830-41.143Q48.092-40.881 48.235-40.545Q48.377-40.209 48.377-39.822Q48.377-39.330 48.113-38.920Q47.850-38.510 47.420-38.279Q46.990-38.049 46.498-38.049Q46.006-38.049 45.572-38.281Q45.139-38.514 44.877-38.922Q44.615-39.330 44.615-39.822M46.498-38.326Q46.955-38.326 47.207-38.549Q47.459-38.772 47.547-39.123Q47.635-39.475 47.635-39.920Q47.635-40.350 47.541-40.688Q47.447-41.025 47.194-41.232Q46.940-41.440 46.498-41.440Q45.850-41.440 45.606-41.023Q45.361-40.607 45.361-39.920Q45.361-39.475 45.449-39.123Q45.537-38.772 45.789-38.549Q46.041-38.326 46.498-38.326",[2846],[2830,14250,14251],{"transform":14244},[2835,14252],{"d":14253,"fill":2832,"stroke":2832,"className":14254,"style":4032},"M50.924-38.049Q50.443-38.049 50.035-38.293Q49.627-38.537 49.389-38.951Q49.150-39.365 49.150-39.854Q49.150-40.346 49.408-40.762Q49.666-41.178 50.098-41.416Q50.529-41.654 51.021-41.654Q51.642-41.654 52.092-41.217L52.092-42.846Q52.092-43.061 52.029-43.156Q51.967-43.252 51.849-43.273Q51.732-43.295 51.486-43.295L51.486-43.592L52.709-43.678L52.709-38.869Q52.709-38.658 52.771-38.563Q52.834-38.467 52.951-38.445Q53.068-38.424 53.318-38.424L53.318-38.127L52.068-38.049L52.068-38.533Q51.603-38.049 50.924-38.049M50.990-38.303Q51.330-38.303 51.623-38.494Q51.916-38.686 52.068-38.982L52.068-40.815Q51.920-41.088 51.658-41.244Q51.396-41.400 51.084-41.400Q50.459-41.400 50.176-40.953Q49.892-40.506 49.892-39.846Q49.892-39.201 50.144-38.752Q50.396-38.303 50.990-38.303",[2846],[2830,14256,14257],{"transform":14244},[2835,14258],{"d":14259,"fill":2832,"stroke":2832,"className":14260,"style":4032},"M54.856-38.760Q55.047-38.486 55.403-38.359Q55.758-38.232 56.141-38.232Q56.477-38.232 56.686-38.418Q56.895-38.604 56.991-38.897Q57.086-39.190 57.086-39.502Q57.086-39.826 56.989-40.121Q56.891-40.416 56.678-40.600Q56.465-40.783 56.133-40.783L55.567-40.783Q55.536-40.783 55.506-40.813Q55.477-40.842 55.477-40.869L55.477-40.951Q55.477-40.986 55.506-41.012Q55.536-41.037 55.567-41.037L56.047-41.072Q56.333-41.072 56.530-41.277Q56.727-41.482 56.823-41.777Q56.918-42.072 56.918-42.350Q56.918-42.729 56.719-42.967Q56.520-43.205 56.141-43.205Q55.821-43.205 55.532-43.098Q55.243-42.990 55.079-42.768Q55.258-42.768 55.381-42.641Q55.504-42.514 55.504-42.342Q55.504-42.170 55.379-42.045Q55.254-41.920 55.079-41.920Q54.907-41.920 54.782-42.045Q54.657-42.170 54.657-42.342Q54.657-42.709 54.881-42.957Q55.106-43.205 55.446-43.326Q55.786-43.447 56.141-43.447Q56.489-43.447 56.852-43.326Q57.215-43.205 57.463-42.955Q57.711-42.705 57.711-42.350Q57.711-41.865 57.393-41.482Q57.075-41.100 56.598-40.928Q57.149-40.818 57.549-40.432Q57.950-40.045 57.950-39.510Q57.950-39.053 57.686-38.697Q57.422-38.342 57.001-38.150Q56.579-37.959 56.141-37.959Q55.731-37.959 55.338-38.094Q54.946-38.229 54.680-38.514Q54.415-38.799 54.415-39.217Q54.415-39.412 54.547-39.541Q54.680-39.670 54.872-39.670Q54.997-39.670 55.100-39.611Q55.204-39.553 55.266-39.447Q55.329-39.342 55.329-39.217Q55.329-39.022 55.194-38.891Q55.059-38.760 54.856-38.760",[2846],[2830,14262,14263,14269,14274],{"stroke":2837,"fontFamily":4023,"fontSize":4024},[2830,14264,14266],{"transform":14265},"translate(74.615 -17.14)",[2835,14267],{"d":14247,"fill":2832,"stroke":2832,"className":14268,"style":4032},[2846],[2830,14270,14271],{"transform":14265},[2835,14272],{"d":14253,"fill":2832,"stroke":2832,"className":14273,"style":4032},[2846],[2830,14275,14276],{"transform":14265},[2835,14277],{"d":14278,"fill":2832,"stroke":2832,"className":14279,"style":4032},"M54.903-39.006L54.840-39.006Q54.981-38.654 55.305-38.443Q55.629-38.232 56.016-38.232Q56.610-38.232 56.860-38.666Q57.110-39.100 57.110-39.736Q57.110-40.330 56.940-40.777Q56.770-41.225 56.270-41.225Q55.973-41.225 55.768-41.145Q55.563-41.065 55.461-40.973Q55.360-40.881 55.245-40.748Q55.129-40.615 55.079-40.600L55.008-40.600Q54.922-40.623 54.903-40.701L54.903-43.350Q54.934-43.447 55.008-43.447Q55.024-43.447 55.032-43.445Q55.040-43.443 55.047-43.440Q55.633-43.190 56.231-43.190Q56.813-43.190 57.430-43.447L57.454-43.447Q57.497-43.447 57.524-43.422Q57.551-43.397 57.551-43.357L57.551-43.279Q57.551-43.248 57.528-43.225Q57.231-42.873 56.809-42.676Q56.387-42.479 55.926-42.479Q55.579-42.479 55.200-42.584L55.200-41.088Q55.418-41.283 55.694-41.381Q55.969-41.479 56.270-41.479Q56.727-41.479 57.096-41.231Q57.465-40.982 57.672-40.578Q57.879-40.174 57.879-39.729Q57.879-39.240 57.624-38.832Q57.368-38.424 56.936-38.191Q56.504-37.959 56.016-37.959Q55.622-37.959 55.266-38.150Q54.911-38.342 54.700-38.676Q54.489-39.010 54.489-39.424Q54.489-39.604 54.606-39.717Q54.723-39.830 54.903-39.830Q55.020-39.830 55.112-39.777Q55.204-39.725 55.256-39.633Q55.309-39.541 55.309-39.424Q55.309-39.240 55.196-39.123Q55.083-39.006 54.903-39.006",[2846],[2835,14281],{"fill":2837,"d":14282,"style":2986},"M-59.912-48.085h202.014",[2830,14284,14285,14292,14298,14304,14310,14316,14322,14328,14334,14340],{"stroke":2837},[2830,14286,14288],{"transform":14287},"translate(-93.206 2.757)",[2835,14289],{"d":14290,"fill":2832,"stroke":2832,"className":14291,"style":2847},"M38.449-38.026Q38.053-38.026 37.767-38.230Q37.482-38.435 37.335-38.769Q37.187-39.103 37.187-39.494Q37.187-39.929 37.361-40.390Q37.535-40.852 37.847-41.243Q38.159-41.634 38.569-41.869Q38.980-42.104 39.420-42.104Q39.688-42.104 39.905-41.966Q40.123-41.827 40.255-41.581Q40.294-41.731 40.402-41.827Q40.510-41.924 40.650-41.924Q40.773-41.924 40.857-41.851Q40.940-41.779 40.940-41.656Q40.940-41.603 40.931-41.572L40.312-39.081Q40.255-38.883 40.255-38.685Q40.255-38.290 40.518-38.290Q40.804-38.290 40.938-38.613Q41.072-38.936 41.191-39.441Q41.200-39.472 41.224-39.496Q41.248-39.520 41.283-39.520L41.389-39.520Q41.437-39.520 41.459-39.487Q41.481-39.454 41.481-39.406Q41.367-38.975 41.276-38.722Q41.186-38.470 40.993-38.248Q40.800-38.026 40.501-38.026Q40.193-38.026 39.945-38.197Q39.697-38.369 39.626-38.659Q39.371-38.373 39.075-38.200Q38.778-38.026 38.449-38.026M38.466-38.290Q38.796-38.290 39.106-38.531Q39.415-38.773 39.626-39.089Q39.635-39.098 39.635-39.116L40.132-41.080Q40.075-41.397 39.883-41.621Q39.692-41.845 39.402-41.845Q39.033-41.845 38.734-41.526Q38.435-41.208 38.268-40.799Q38.132-40.452 38.007-39.942Q37.882-39.432 37.882-39.107Q37.882-38.782 38.020-38.536Q38.159-38.290 38.466-38.290",[2846],[2830,14293,14294],{"transform":14287},[2835,14295],{"d":14296,"fill":2832,"stroke":2832,"className":14297,"style":6553},"M44.768-37.127L42.477-37.127L42.477-37.385Q43.353-37.385 43.353-37.558L43.353-40.637Q43.160-40.549 42.928-40.512Q42.697-40.476 42.442-40.476L42.442-40.733Q42.820-40.733 43.141-40.818Q43.461-40.903 43.690-41.117L43.810-41.117Q43.842-41.117 43.867-41.094Q43.892-41.070 43.892-41.032L43.892-37.558Q43.892-37.385 44.768-37.385",[2846],[2830,14299,14300],{"transform":14287},[2835,14301],{"d":14302,"fill":2832,"stroke":2832,"className":14303,"style":2847},"M48.315-38.127L46.408-38.127Q46.369-38.127 46.338-38.167Q46.307-38.206 46.307-38.246Q46.307-38.307 46.340-38.375Q46.373-38.443 46.435-38.443Q46.817-38.443 47.070-38.571Q47.322-38.698 47.415-39.019L48.583-43.690Q48.601-43.778 48.601-43.822Q48.601-43.888 48.575-43.906Q48.403-43.959 47.876-43.959Q47.766-43.959 47.766-44.077Q47.766-44.139 47.799-44.207Q47.832-44.275 47.894-44.275L49.476-44.275Q49.528-44.275 49.568-44.246Q49.607-44.218 49.612-44.165L50.350-39.019L53.659-44.165Q53.738-44.275 53.857-44.275L55.382-44.275Q55.426-44.275 55.452-44.242Q55.478-44.209 55.478-44.165Q55.478-44.086 55.448-44.022Q55.417-43.959 55.351-43.959Q54.859-43.959 54.635-43.906Q54.485-43.853 54.437-43.625L53.211-38.711Q53.193-38.668 53.193-38.575Q53.193-38.514 53.220-38.496Q53.391-38.443 53.923-38.443Q53.967-38.443 53.993-38.410Q54.019-38.377 54.019-38.334Q54.019-38.127 53.857-38.127L51.537-38.127Q51.440-38.127 51.440-38.246Q51.440-38.307 51.473-38.375Q51.506-38.443 51.563-38.443Q52.068-38.443 52.292-38.496Q52.420-38.544 52.490-38.773L53.778-43.932L50.117-38.246Q50.042-38.127 49.906-38.127Q49.779-38.127 49.766-38.246L48.953-43.871L47.722-38.953Q47.705-38.865 47.705-38.813Q47.705-38.584 47.898-38.514Q48.091-38.443 48.377-38.443Q48.421-38.443 48.449-38.410Q48.478-38.377 48.478-38.334Q48.478-38.127 48.315-38.127",[2846],[2830,14305,14306],{"transform":14287},[2835,14307],{"d":14308,"fill":2832,"stroke":2832,"className":14309,"style":6553},"M57.871-37.127L55.580-37.127L55.580-37.385Q56.456-37.385 56.456-37.558L56.456-40.637Q56.263-40.549 56.031-40.512Q55.800-40.476 55.545-40.476L55.545-40.733Q55.923-40.733 56.244-40.818Q56.564-40.903 56.793-41.117L56.913-41.117Q56.945-41.117 56.970-41.094Q56.995-41.070 56.995-41.032L56.995-37.558Q56.995-37.385 57.871-37.385",[2846],[2830,14311,14312],{"transform":14287},[2835,14313],{"d":14314,"fill":2832,"stroke":2832,"className":14315,"style":2847},"M61.417-38.127L59.510-38.127Q59.471-38.127 59.440-38.167Q59.409-38.206 59.409-38.246Q59.409-38.307 59.442-38.375Q59.475-38.443 59.537-38.443Q59.919-38.443 60.172-38.571Q60.424-38.698 60.517-39.019L61.685-43.690Q61.703-43.778 61.703-43.822Q61.703-43.888 61.677-43.906Q61.505-43.959 60.978-43.959Q60.868-43.959 60.868-44.077Q60.868-44.139 60.901-44.207Q60.934-44.275 60.996-44.275L62.578-44.275Q62.630-44.275 62.670-44.246Q62.709-44.218 62.714-44.165L63.452-39.019L66.761-44.165Q66.840-44.275 66.959-44.275L68.484-44.275Q68.528-44.275 68.554-44.242Q68.580-44.209 68.580-44.165Q68.580-44.086 68.550-44.022Q68.519-43.959 68.453-43.959Q67.961-43.959 67.737-43.906Q67.587-43.853 67.539-43.625L66.313-38.711Q66.295-38.668 66.295-38.575Q66.295-38.514 66.322-38.496Q66.493-38.443 67.025-38.443Q67.069-38.443 67.095-38.410Q67.121-38.377 67.121-38.334Q67.121-38.127 66.959-38.127L64.639-38.127Q64.542-38.127 64.542-38.246Q64.542-38.307 64.575-38.375Q64.608-38.443 64.665-38.443Q65.170-38.443 65.394-38.496Q65.522-38.544 65.592-38.773L66.880-43.932L63.219-38.246Q63.144-38.127 63.008-38.127Q62.881-38.127 62.868-38.246L62.055-43.871L60.824-38.953Q60.807-38.865 60.807-38.813Q60.807-38.584 61-38.514Q61.193-38.443 61.479-38.443Q61.523-38.443 61.551-38.410Q61.580-38.377 61.580-38.334Q61.580-38.127 61.417-38.127",[2846],[2830,14317,14318],{"transform":14287},[2835,14319],{"d":14320,"fill":2832,"stroke":2832,"className":14321,"style":6553},"M73.783-43.405L69.799-43.405Q69.740-43.414 69.697-43.456Q69.655-43.498 69.655-43.560Q69.655-43.622 69.697-43.664Q69.740-43.706 69.799-43.715L73.783-43.715Q73.844-43.706 73.885-43.665Q73.926-43.624 73.926-43.560Q73.926-43.496 73.885-43.455Q73.844-43.414 73.783-43.405",[2846],[2830,14323,14324],{"transform":14287},[2835,14325],{"d":14326,"fill":2832,"stroke":2832,"className":14327,"style":6553},"M77.723-42.060L75.432-42.060L75.432-42.318Q76.308-42.318 76.308-42.491L76.308-45.570Q76.115-45.482 75.883-45.445Q75.652-45.409 75.397-45.409L75.397-45.666Q75.775-45.666 76.096-45.751Q76.416-45.836 76.645-46.050L76.765-46.050Q76.797-46.050 76.822-46.027Q76.847-46.003 76.847-45.965L76.847-42.491Q76.847-42.318 77.723-42.318",[2846],[2830,14329,14330],{"transform":14287},[2835,14331],{"d":14332,"fill":2832,"stroke":2832,"className":14333,"style":6553},"M70.973-35.840L68.682-35.840L68.682-36.098Q69.558-36.098 69.558-36.271L69.558-39.350Q69.365-39.262 69.133-39.225Q68.902-39.189 68.647-39.189L68.647-39.446Q69.025-39.446 69.346-39.531Q69.666-39.616 69.895-39.830L70.015-39.830Q70.047-39.830 70.072-39.807Q70.097-39.783 70.097-39.745L70.097-36.271Q70.097-36.098 70.973-36.098",[2846],[2830,14335,14336],{"transform":14287},[2835,14337],{"d":14338,"fill":2832,"stroke":2832,"className":14339,"style":2847},"M87.918-39.270L82.112-39.270Q82.033-39.283 81.983-39.333Q81.932-39.384 81.932-39.459Q81.932-39.608 82.112-39.656L87.918-39.656Q88.089-39.604 88.089-39.459Q88.089-39.305 87.918-39.270M87.918-41.098L82.112-41.098Q81.932-41.128 81.932-41.287Q81.932-41.436 82.112-41.484L87.918-41.484Q88.089-41.432 88.089-41.287Q88.089-41.133 87.918-41.098",[2846],[2830,14341,14342],{"transform":14287},[2835,14343],{"d":14344,"fill":2832,"stroke":2832,"className":14345,"style":2847},"M95.089-38.127L91.639-38.127L91.639-38.360Q91.639-38.373 91.670-38.404L93.124-39.981Q93.590-40.478 93.843-40.783Q94.096-41.089 94.287-41.500Q94.478-41.911 94.478-42.350Q94.478-42.939 94.155-43.372Q93.832-43.805 93.252-43.805Q92.988-43.805 92.742-43.695Q92.496-43.585 92.320-43.398Q92.144-43.211 92.048-42.961L92.127-42.961Q92.329-42.961 92.472-42.825Q92.615-42.689 92.615-42.473Q92.615-42.267 92.472-42.128Q92.329-41.990 92.127-41.990Q91.925-41.990 91.782-42.133Q91.639-42.275 91.639-42.473Q91.639-42.935 91.876-43.308Q92.114-43.682 92.514-43.901Q92.913-44.121 93.362-44.121Q93.885-44.121 94.339-43.906Q94.794-43.690 95.067-43.291Q95.339-42.891 95.339-42.350Q95.339-41.955 95.168-41.601Q94.996-41.247 94.731-40.968Q94.465-40.689 94.014-40.304Q93.564-39.920 93.485-39.845L92.461-38.883L93.278-38.883Q93.929-38.883 94.366-38.894Q94.803-38.905 94.834-38.927Q94.904-39.010 94.959-39.250Q95.014-39.489 95.054-39.757L95.339-39.757L95.089-38.127M98.112-37.929Q96.987-37.929 96.574-38.826Q96.161-39.722 96.161-40.997Q96.161-41.770 96.310-42.469Q96.460-43.168 96.895-43.644Q97.330-44.121 98.112-44.121Q98.890-44.121 99.325-43.642Q99.760-43.163 99.910-42.467Q100.059-41.770 100.059-40.997Q100.059-39.718 99.646-38.824Q99.233-37.929 98.112-37.929M98.112-38.189Q98.631-38.189 98.881-38.700Q99.132-39.212 99.189-39.823Q99.246-40.434 99.246-41.142Q99.246-41.827 99.189-42.387Q99.132-42.948 98.879-43.405Q98.626-43.862 98.112-43.862Q97.708-43.862 97.471-43.585Q97.233-43.308 97.126-42.867Q97.018-42.425 96.994-42.032Q96.970-41.638 96.970-41.142Q96.970-40.636 96.994-40.208Q97.018-39.779 97.126-39.296Q97.233-38.813 97.473-38.501Q97.712-38.189 98.112-38.189",[2846],[2830,14347,14348],{"fill":2953,"stroke":2953},[2830,14349,14350],{"transform":3935},[2835,14351],{"d":14352,"fill":2953,"stroke":2953,"className":14353,"style":2847},"M40.734-38.127L37.284-38.127L37.284-38.360Q37.284-38.373 37.315-38.404L38.769-39.981Q39.235-40.478 39.488-40.783Q39.741-41.089 39.932-41.500Q40.123-41.911 40.123-42.350Q40.123-42.939 39.800-43.372Q39.477-43.805 38.897-43.805Q38.633-43.805 38.387-43.695Q38.141-43.585 37.965-43.398Q37.789-43.211 37.693-42.961L37.772-42.961Q37.974-42.961 38.117-42.825Q38.260-42.689 38.260-42.473Q38.260-42.267 38.117-42.128Q37.974-41.990 37.772-41.990Q37.570-41.990 37.427-42.133Q37.284-42.275 37.284-42.473Q37.284-42.935 37.521-43.308Q37.759-43.682 38.159-43.901Q38.558-44.121 39.007-44.121Q39.530-44.121 39.984-43.906Q40.439-43.690 40.712-43.291Q40.984-42.891 40.984-42.350Q40.984-41.955 40.813-41.601Q40.641-41.247 40.376-40.968Q40.110-40.689 39.659-40.304Q39.209-39.920 39.130-39.845L38.106-38.883L38.923-38.883Q39.574-38.883 40.011-38.894Q40.448-38.905 40.479-38.927Q40.549-39.010 40.604-39.250Q40.659-39.489 40.699-39.757L40.984-39.757",[2846],[2830,14355,14357],{"fill":14356,"stroke":14356},"var(--tk-line)",[2830,14358,14360],{"transform":14359},"translate(83.046 2.9)",[2835,14361],{"d":14362,"fill":14356,"stroke":14356,"className":14363,"style":2847},"M39.139-37.929Q38.014-37.929 37.600-38.826Q37.187-39.722 37.187-40.997Q37.187-41.770 37.337-42.469Q37.486-43.168 37.921-43.644Q38.356-44.121 39.139-44.121Q39.916-44.121 40.351-43.642Q40.786-43.163 40.936-42.467Q41.085-41.770 41.085-40.997Q41.085-39.718 40.672-38.824Q40.259-37.929 39.139-37.929M39.139-38.189Q39.657-38.189 39.908-38.700Q40.158-39.212 40.215-39.823Q40.272-40.434 40.272-41.142Q40.272-41.827 40.215-42.387Q40.158-42.948 39.905-43.405Q39.653-43.862 39.139-43.862Q38.734-43.862 38.497-43.585Q38.260-43.308 38.152-42.867Q38.044-42.425 38.020-42.032Q37.996-41.638 37.996-41.142Q37.996-40.636 38.020-40.208Q38.044-39.779 38.152-39.296Q38.260-38.813 38.499-38.501Q38.739-38.189 39.139-38.189",[2846],[2830,14365,14366,14372,14378,14383,14389,14394,14399,14404,14410,14415],{"stroke":2837},[2830,14367,14369],{"transform":14368},"translate(-93.206 28.364)",[2835,14370],{"d":14290,"fill":2832,"stroke":2832,"className":14371,"style":2847},[2846],[2830,14373,14374],{"transform":14368},[2835,14375],{"d":14376,"fill":2832,"stroke":2832,"className":14377,"style":6553},"M44.768-37.127L42.158-37.127L42.158-37.312Q42.164-37.335 42.184-37.361L43.335-38.416Q43.675-38.727 43.855-38.913Q44.036-39.099 44.181-39.359Q44.326-39.620 44.326-39.916Q44.326-40.189 44.200-40.404Q44.074-40.619 43.854-40.739Q43.634-40.859 43.359-40.859Q43.183-40.859 43.013-40.802Q42.843-40.745 42.711-40.638Q42.580-40.531 42.500-40.373Q42.588-40.373 42.666-40.329Q42.744-40.285 42.788-40.209Q42.831-40.133 42.831-40.036Q42.831-39.896 42.735-39.799Q42.638-39.702 42.495-39.702Q42.357-39.702 42.257-39.802Q42.158-39.901 42.158-40.036Q42.158-40.361 42.348-40.609Q42.539-40.856 42.842-40.987Q43.145-41.117 43.461-41.117Q43.842-41.117 44.185-40.982Q44.528-40.848 44.742-40.575Q44.956-40.303 44.956-39.916Q44.956-39.641 44.831-39.414Q44.706-39.187 44.526-39.015Q44.346-38.844 44.021-38.604Q43.696-38.363 43.611-38.296L42.855-37.692L43.388-37.692Q43.877-37.692 44.208-37.700Q44.539-37.707 44.554-37.722Q44.613-37.792 44.645-37.927Q44.677-38.062 44.709-38.273L44.956-38.273",[2846],[2830,14379,14380],{"transform":14368},[2835,14381],{"d":14302,"fill":2832,"stroke":2832,"className":14382,"style":2847},[2846],[2830,14384,14385],{"transform":14368},[2835,14386],{"d":14387,"fill":2832,"stroke":2832,"className":14388,"style":6553},"M57.871-37.127L55.261-37.127L55.261-37.312Q55.267-37.335 55.287-37.361L56.438-38.416Q56.778-38.727 56.958-38.913Q57.139-39.099 57.284-39.359Q57.429-39.620 57.429-39.916Q57.429-40.189 57.303-40.404Q57.177-40.619 56.957-40.739Q56.737-40.859 56.462-40.859Q56.286-40.859 56.116-40.802Q55.946-40.745 55.814-40.638Q55.683-40.531 55.603-40.373Q55.691-40.373 55.769-40.329Q55.847-40.285 55.891-40.209Q55.934-40.133 55.934-40.036Q55.934-39.896 55.838-39.799Q55.741-39.702 55.598-39.702Q55.460-39.702 55.360-39.802Q55.261-39.901 55.261-40.036Q55.261-40.361 55.451-40.609Q55.642-40.856 55.945-40.987Q56.248-41.117 56.564-41.117Q56.945-41.117 57.288-40.982Q57.631-40.848 57.845-40.575Q58.059-40.303 58.059-39.916Q58.059-39.641 57.934-39.414Q57.809-39.187 57.629-39.015Q57.449-38.844 57.124-38.604Q56.799-38.363 56.714-38.296L55.958-37.692L56.491-37.692Q56.980-37.692 57.311-37.700Q57.642-37.707 57.657-37.722Q57.716-37.792 57.748-37.927Q57.780-38.062 57.812-38.273L58.059-38.273",[2846],[2830,14390,14391],{"transform":14368},[2835,14392],{"d":14314,"fill":2832,"stroke":2832,"className":14393,"style":2847},[2846],[2830,14395,14396],{"transform":14368},[2835,14397],{"d":14320,"fill":2832,"stroke":2832,"className":14398,"style":6553},[2846],[2830,14400,14401],{"transform":14368},[2835,14402],{"d":14326,"fill":2832,"stroke":2832,"className":14403,"style":6553},[2846],[2830,14405,14406],{"transform":14368},[2835,14407],{"d":14408,"fill":2832,"stroke":2832,"className":14409,"style":6553},"M70.973-35.840L68.363-35.840L68.363-36.025Q68.369-36.048 68.389-36.074L69.540-37.129Q69.880-37.440 70.060-37.626Q70.241-37.812 70.386-38.072Q70.531-38.333 70.531-38.629Q70.531-38.902 70.405-39.117Q70.279-39.332 70.059-39.452Q69.839-39.572 69.564-39.572Q69.388-39.572 69.218-39.515Q69.048-39.458 68.916-39.351Q68.785-39.244 68.705-39.086Q68.793-39.086 68.871-39.042Q68.949-38.998 68.993-38.922Q69.036-38.846 69.036-38.749Q69.036-38.609 68.940-38.512Q68.843-38.415 68.700-38.415Q68.562-38.415 68.462-38.515Q68.363-38.614 68.363-38.749Q68.363-39.074 68.553-39.322Q68.744-39.569 69.047-39.700Q69.350-39.830 69.666-39.830Q70.047-39.830 70.390-39.695Q70.733-39.561 70.947-39.288Q71.161-39.016 71.161-38.629Q71.161-38.354 71.036-38.127Q70.911-37.900 70.731-37.728Q70.551-37.557 70.226-37.317Q69.901-37.076 69.816-37.009L69.060-36.405L69.593-36.405Q70.082-36.405 70.413-36.413Q70.745-36.420 70.759-36.435Q70.818-36.505 70.850-36.640Q70.882-36.775 70.914-36.986L71.161-36.986",[2846],[2830,14411,14412],{"transform":14368},[2835,14413],{"d":14338,"fill":2832,"stroke":2832,"className":14414,"style":2847},[2846],[2830,14416,14417],{"transform":14368},[2835,14418],{"d":14419,"fill":2832,"stroke":2832,"className":14420,"style":2847},"M95.089-38.127L92.057-38.127L92.057-38.443Q93.208-38.443 93.208-38.738L93.208-43.462Q92.720-43.229 91.999-43.229L91.999-43.545Q93.129-43.545 93.691-44.121L93.836-44.121Q93.871-44.121 93.904-44.088Q93.937-44.055 93.937-44.020L93.937-38.738Q93.937-38.443 95.089-38.443L95.089-38.127M96.187-39.494Q96.187-40.052 96.548-40.465Q96.908-40.878 97.484-41.150L97.115-41.383Q96.811-41.585 96.625-41.915Q96.438-42.245 96.438-42.601Q96.438-43.255 96.943-43.688Q97.449-44.121 98.112-44.121Q98.512-44.121 98.897-43.961Q99.281-43.800 99.529-43.495Q99.778-43.190 99.778-42.772Q99.778-41.941 98.710-41.383L99.264-41.036Q99.611-40.808 99.822-40.439Q100.033-40.069 100.033-39.656Q100.033-39.278 99.874-38.960Q99.716-38.641 99.439-38.408Q99.162-38.175 98.820-38.052Q98.477-37.929 98.112-37.929Q97.646-37.929 97.200-38.116Q96.754-38.303 96.471-38.657Q96.187-39.010 96.187-39.494M96.710-39.494Q96.710-38.949 97.130-38.582Q97.550-38.215 98.112-38.215Q98.442-38.215 98.767-38.347Q99.092-38.479 99.301-38.733Q99.510-38.988 99.510-39.331Q99.510-39.595 99.373-39.819Q99.237-40.043 99.004-40.197L97.761-40.979Q97.299-40.742 97.005-40.355Q96.710-39.968 96.710-39.494M97.321-42.249L98.437-41.546Q98.661-41.669 98.866-41.858Q99.070-42.047 99.191-42.280Q99.312-42.513 99.312-42.772Q99.312-43.080 99.140-43.330Q98.969-43.581 98.692-43.721Q98.415-43.862 98.103-43.862Q97.655-43.862 97.282-43.616Q96.908-43.370 96.908-42.943Q96.908-42.539 97.321-42.249",[2846],[2830,14422,14423],{"fill":14356,"stroke":14356},[2830,14424,14426],{"transform":14425},"translate(37.521 28.507)",[2835,14427],{"d":14362,"fill":14356,"stroke":14356,"className":14428,"style":2847},[2846],[2830,14430,14431],{"fill":2953,"stroke":2953},[2830,14432,14434],{"transform":14433},"translate(83.046 28.507)",[2835,14435],{"d":14436,"fill":2953,"stroke":2953,"className":14437,"style":2847},"M37.728-38.848L37.684-38.848Q37.886-38.531 38.273-38.373Q38.660-38.215 39.086-38.215Q39.622-38.215 39.861-38.650Q40.101-39.085 40.101-39.665Q40.101-40.245 39.855-40.685Q39.609-41.124 39.077-41.124L38.457-41.124Q38.431-41.124 38.398-41.153Q38.365-41.181 38.365-41.203L38.365-41.304Q38.365-41.335 38.394-41.359Q38.422-41.383 38.457-41.383L38.976-41.423Q39.442-41.423 39.688-41.895Q39.934-42.368 39.934-42.886Q39.934-43.313 39.721-43.587Q39.508-43.862 39.086-43.862Q38.743-43.862 38.418-43.732Q38.093-43.603 37.908-43.348L37.934-43.348Q38.137-43.348 38.273-43.207Q38.409-43.066 38.409-42.869Q38.409-42.671 38.275-42.537Q38.141-42.403 37.943-42.403Q37.741-42.403 37.603-42.537Q37.464-42.671 37.464-42.869Q37.464-43.458 37.967-43.789Q38.471-44.121 39.086-44.121Q39.464-44.121 39.866-43.981Q40.268-43.840 40.536-43.561Q40.804-43.282 40.804-42.886Q40.804-42.337 40.450-41.900Q40.097-41.462 39.556-41.278Q39.947-41.199 40.292-40.975Q40.637-40.751 40.848-40.410Q41.059-40.069 41.059-39.674Q41.059-39.292 40.896-38.969Q40.734-38.646 40.442-38.410Q40.149-38.175 39.802-38.052Q39.455-37.929 39.086-37.929Q38.638-37.929 38.207-38.090Q37.776-38.250 37.495-38.577Q37.214-38.905 37.214-39.362Q37.214-39.577 37.361-39.720Q37.508-39.863 37.728-39.863Q37.939-39.863 38.084-39.718Q38.229-39.573 38.229-39.362Q38.229-39.151 38.082-38.999Q37.934-38.848 37.728-38.848",[2846],[2835,14439],{"fill":2837,"d":14440},"M-59.912.285h202.014",[2830,14442,14443,14450,14456,14462,14468,14474],{"stroke":2837,"fontSize":7134},[2830,14444,14446],{"transform":14445},"translate(-93.206 53.465)",[2835,14447],{"d":14448,"fill":2832,"stroke":2832,"className":14449,"style":2847},"M37.135-38.109L37.135-39.551Q37.135-39.582 37.163-39.606Q37.192-39.630 37.223-39.630L37.332-39.630Q37.368-39.630 37.389-39.608Q37.411-39.586 37.420-39.551Q37.680-38.290 38.646-38.290Q39.073-38.290 39.367-38.474Q39.661-38.659 39.661-39.063Q39.661-39.357 39.431-39.553Q39.200-39.749 38.888-39.810L38.286-39.929Q37.820-40.017 37.477-40.300Q37.135-40.584 37.135-41.023Q37.135-41.616 37.572-41.889Q38.009-42.161 38.646-42.161Q39.125-42.161 39.473-41.915L39.723-42.139Q39.780-42.161 39.780-42.161L39.833-42.161Q39.859-42.161 39.892-42.135Q39.925-42.108 39.925-42.078L39.925-40.918Q39.925-40.887 39.890-40.860Q39.855-40.834 39.833-40.834L39.723-40.834Q39.701-40.834 39.668-40.863Q39.635-40.891 39.635-40.918Q39.635-41.383 39.369-41.654Q39.103-41.924 38.638-41.924Q38.233-41.924 37.930-41.779Q37.627-41.634 37.627-41.278Q37.627-41.032 37.844-40.878Q38.062-40.724 38.339-40.667L38.967-40.540Q39.284-40.478 39.554-40.311Q39.824-40.144 39.991-39.878Q40.158-39.612 40.158-39.296Q40.158-38.654 39.732-38.340Q39.306-38.026 38.646-38.026Q38.374-38.026 38.108-38.120Q37.842-38.215 37.662-38.404L37.341-38.065Q37.324-38.026 37.275-38.026L37.223-38.026Q37.201-38.026 37.168-38.054Q37.135-38.083 37.135-38.109M41.459-39.199L41.459-41.186Q41.459-41.427 41.389-41.535Q41.318-41.643 41.184-41.667Q41.050-41.691 40.769-41.691L40.769-42.007L42.162-42.104L42.162-39.234Q42.162-38.856 42.208-38.670Q42.254-38.483 42.426-38.386Q42.597-38.290 42.953-38.290Q43.287-38.290 43.526-38.481Q43.766-38.672 43.891-38.975Q44.016-39.278 44.016-39.595L44.016-41.186Q44.016-41.427 43.946-41.535Q43.876-41.643 43.742-41.667Q43.608-41.691 43.322-41.691L43.322-42.007L44.720-42.104L44.720-38.944Q44.720-38.707 44.790-38.599Q44.860-38.492 44.994-38.468Q45.128-38.443 45.410-38.443L45.410-38.127L44.043-38.026L44.043-38.747Q43.876-38.421 43.570-38.224Q43.265-38.026 42.909-38.026Q42.250-38.026 41.854-38.296Q41.459-38.566 41.459-39.199M47.994-38.127L45.906-38.127L45.906-38.443Q46.214-38.443 46.405-38.496Q46.596-38.549 46.596-38.738L46.596-41.186Q46.596-41.427 46.526-41.535Q46.455-41.643 46.321-41.667Q46.187-41.691 45.906-41.691L45.906-42.007L47.246-42.104L47.246-41.269Q47.444-41.647 47.805-41.876Q48.165-42.104 48.587-42.104Q49.633-42.104 49.817-41.295Q50.019-41.665 50.378-41.884Q50.736-42.104 51.153-42.104Q51.777-42.104 52.102-41.810Q52.428-41.515 52.428-40.891L52.428-38.738Q52.428-38.549 52.621-38.496Q52.814-38.443 53.122-38.443L53.122-38.127L51.035-38.127L51.035-38.443Q51.342-38.443 51.535-38.496Q51.729-38.549 51.729-38.738L51.729-40.856Q51.729-41.287 51.601-41.566Q51.474-41.845 51.087-41.845Q50.744-41.845 50.461-41.656Q50.178-41.467 50.022-41.155Q49.866-40.843 49.866-40.496L49.866-38.738Q49.866-38.549 50.057-38.496Q50.248-38.443 50.556-38.443L50.556-38.127L48.468-38.127L48.468-38.443Q48.780-38.443 48.971-38.496Q49.162-38.549 49.162-38.738L49.162-40.856Q49.162-41.115 49.119-41.337Q49.075-41.559 48.930-41.702Q48.785-41.845 48.525-41.845Q47.989-41.845 47.644-41.438Q47.299-41.032 47.299-40.496L47.299-38.738Q47.299-38.549 47.493-38.496Q47.686-38.443 47.994-38.443",[2846],[2830,14451,14452],{"transform":14445},[2835,14453],{"d":14454,"fill":2832,"stroke":2832,"className":14455,"style":2847},"M62.906-39.270L57.100-39.270Q57.021-39.283 56.971-39.333Q56.920-39.384 56.920-39.459Q56.920-39.608 57.100-39.656L62.906-39.656Q63.077-39.604 63.077-39.459Q63.077-39.305 62.906-39.270M62.906-41.098L57.100-41.098Q56.920-41.128 56.920-41.287Q56.920-41.436 57.100-41.484L62.906-41.484Q63.077-41.432 63.077-41.287Q63.077-41.133 62.906-41.098",[2846],[2830,14457,14458],{"transform":14445},[2835,14459],{"d":14460,"fill":2832,"stroke":2832,"className":14461,"style":2847},"M67.071-38.848L67.027-38.848Q67.229-38.531 67.616-38.373Q68.003-38.215 68.429-38.215Q68.965-38.215 69.204-38.650Q69.444-39.085 69.444-39.665Q69.444-40.245 69.198-40.685Q68.952-41.124 68.420-41.124L67.800-41.124Q67.774-41.124 67.741-41.153Q67.708-41.181 67.708-41.203L67.708-41.304Q67.708-41.335 67.737-41.359Q67.765-41.383 67.800-41.383L68.319-41.423Q68.785-41.423 69.031-41.895Q69.277-42.368 69.277-42.886Q69.277-43.313 69.064-43.587Q68.851-43.862 68.429-43.862Q68.086-43.862 67.761-43.732Q67.436-43.603 67.251-43.348L67.277-43.348Q67.480-43.348 67.616-43.207Q67.752-43.066 67.752-42.869Q67.752-42.671 67.618-42.537Q67.484-42.403 67.286-42.403Q67.084-42.403 66.946-42.537Q66.807-42.671 66.807-42.869Q66.807-43.458 67.310-43.789Q67.814-44.121 68.429-44.121Q68.807-44.121 69.209-43.981Q69.611-43.840 69.879-43.561Q70.147-43.282 70.147-42.886Q70.147-42.337 69.793-41.900Q69.440-41.462 68.899-41.278Q69.290-41.199 69.635-40.975Q69.980-40.751 70.191-40.410Q70.402-40.069 70.402-39.674Q70.402-39.292 70.239-38.969Q70.077-38.646 69.785-38.410Q69.492-38.175 69.145-38.052Q68.798-37.929 68.429-37.929Q67.981-37.929 67.550-38.090Q67.119-38.250 66.838-38.577Q66.557-38.905 66.557-39.362Q66.557-39.577 66.704-39.720Q66.851-39.863 67.071-39.863Q67.282-39.863 67.427-39.718Q67.572-39.573 67.572-39.362Q67.572-39.151 67.425-38.999Q67.277-38.848 67.071-38.848M71.175-39.494Q71.175-40.052 71.536-40.465Q71.896-40.878 72.472-41.150L72.103-41.383Q71.799-41.585 71.613-41.915Q71.426-42.245 71.426-42.601Q71.426-43.255 71.931-43.688Q72.437-44.121 73.100-44.121Q73.500-44.121 73.885-43.961Q74.269-43.800 74.517-43.495Q74.766-43.190 74.766-42.772Q74.766-41.941 73.698-41.383L74.252-41.036Q74.599-40.808 74.810-40.439Q75.021-40.069 75.021-39.656Q75.021-39.278 74.862-38.960Q74.704-38.641 74.427-38.408Q74.150-38.175 73.808-38.052Q73.465-37.929 73.100-37.929Q72.634-37.929 72.188-38.116Q71.742-38.303 71.459-38.657Q71.175-39.010 71.175-39.494M71.698-39.494Q71.698-38.949 72.118-38.582Q72.538-38.215 73.100-38.215Q73.430-38.215 73.755-38.347Q74.080-38.479 74.289-38.733Q74.498-38.988 74.498-39.331Q74.498-39.595 74.361-39.819Q74.225-40.043 73.992-40.197L72.749-40.979Q72.287-40.742 71.993-40.355Q71.698-39.968 71.698-39.494M72.309-42.249L73.425-41.546Q73.649-41.669 73.854-41.858Q74.058-42.047 74.179-42.280Q74.300-42.513 74.300-42.772Q74.300-43.080 74.128-43.330Q73.957-43.581 73.680-43.721Q73.403-43.862 73.091-43.862Q72.643-43.862 72.270-43.616Q71.896-43.370 71.896-42.943Q71.896-42.539 72.309-42.249",[2846],[2830,14463,14464],{"transform":14445},[2835,14465],{"d":14466,"fill":2832,"stroke":2832,"className":14467,"style":2847},"M84.489-38.351L78.683-38.351Q78.600-38.364 78.552-38.421Q78.503-38.479 78.503-38.549Q78.503-38.615 78.554-38.670Q78.604-38.725 78.683-38.738L84.489-38.738Q84.660-38.703 84.660-38.549Q84.660-38.479 84.612-38.421Q84.563-38.364 84.489-38.351M84.489-40.179L78.683-40.179Q78.503-40.210 78.503-40.377Q78.503-40.443 78.554-40.498Q78.604-40.553 78.683-40.566L84.489-40.566Q84.660-40.531 84.660-40.377Q84.660-40.214 84.489-40.179M84.489-42.007L78.683-42.007Q78.503-42.038 78.503-42.205Q78.503-42.275 78.552-42.328Q78.600-42.381 78.683-42.394L84.489-42.394Q84.660-42.359 84.660-42.205Q84.660-42.043 84.489-42.007",[2846],[2830,14469,14470],{"transform":14445},[2835,14471],{"d":14472,"fill":2832,"stroke":2832,"className":14473,"style":2847},"M88.140-39.494Q88.140-40.052 88.500-40.465Q88.860-40.878 89.436-41.150L89.067-41.383Q88.764-41.585 88.577-41.915Q88.390-42.245 88.390-42.601Q88.390-43.255 88.896-43.688Q89.401-44.121 90.065-44.121Q90.464-44.121 90.849-43.961Q91.233-43.800 91.482-43.495Q91.730-43.190 91.730-42.772Q91.730-41.941 90.662-41.383L91.216-41.036Q91.563-40.808 91.774-40.439Q91.985-40.069 91.985-39.656Q91.985-39.278 91.827-38.960Q91.669-38.641 91.392-38.408Q91.115-38.175 90.772-38.052Q90.429-37.929 90.065-37.929Q89.599-37.929 89.153-38.116Q88.707-38.303 88.423-38.657Q88.140-39.010 88.140-39.494M88.663-39.494Q88.663-38.949 89.082-38.582Q89.502-38.215 90.065-38.215Q90.394-38.215 90.719-38.347Q91.045-38.479 91.253-38.733Q91.462-38.988 91.462-39.331Q91.462-39.595 91.326-39.819Q91.190-40.043 90.957-40.197L89.713-40.979Q89.252-40.742 88.957-40.355Q88.663-39.968 88.663-39.494M89.274-42.249L90.390-41.546Q90.614-41.669 90.818-41.858Q91.023-42.047 91.143-42.280Q91.264-42.513 91.264-42.772Q91.264-43.080 91.093-43.330Q90.921-43.581 90.645-43.721Q90.368-43.862 90.056-43.862Q89.607-43.862 89.234-43.616Q88.860-43.370 88.860-42.943Q88.860-42.539 89.274-42.249",[2846],[2830,14475,14476],{"transform":14445},[2835,14477],{"d":14478,"fill":2832,"stroke":2832,"className":14479,"style":2847},"M98.313-35.886Q97.808-36.273 97.439-36.778Q97.069-37.283 96.826-37.883Q96.582-38.483 96.467-39.100Q96.353-39.718 96.353-40.377Q96.353-41.036 96.467-41.651Q96.582-42.267 96.821-42.860Q97.061-43.453 97.434-43.963Q97.808-44.473 98.313-44.859Q98.348-44.877 98.370-44.877L98.449-44.877Q98.537-44.877 98.537-44.776Q98.537-44.741 98.502-44.706Q97.940-44.183 97.595-43.477Q97.250-42.772 97.102-41.990Q96.955-41.208 96.955-40.377Q96.955-39.753 97.034-39.169Q97.113-38.584 97.291-38.017Q97.469-37.450 97.768-36.949Q98.067-36.448 98.502-36.040Q98.537-36.004 98.537-35.965Q98.537-35.868 98.449-35.868L98.370-35.868Q98.348-35.868 98.313-35.886M102.958-38.127L99.926-38.127L99.926-38.443Q101.077-38.443 101.077-38.738L101.077-43.462Q100.589-43.229 99.869-43.229L99.869-43.545Q100.998-43.545 101.561-44.121L101.706-44.121Q101.741-44.121 101.774-44.088Q101.807-44.055 101.807-44.020L101.807-38.738Q101.807-38.443 102.958-38.443L102.958-38.127M104.496-39.133Q104.637-38.720 104.997-38.468Q105.357-38.215 105.793-38.215Q106.245-38.215 106.511-38.468Q106.777-38.720 106.880-39.105Q106.983-39.489 106.983-39.946Q106.983-41.647 106.074-41.647Q105.753-41.647 105.524-41.553Q105.296-41.458 105.166-41.339Q105.037-41.221 104.925-41.082Q104.813-40.944 104.777-40.935L104.694-40.935Q104.650-40.935 104.619-40.966Q104.588-40.997 104.588-41.045L104.588-44.042Q104.588-44.073 104.624-44.097Q104.659-44.121 104.685-44.121L104.725-44.121Q105.357-43.831 106.030-43.831Q106.702-43.831 107.344-44.121L107.370-44.121Q107.401-44.121 107.434-44.099Q107.467-44.077 107.467-44.042L107.467-43.941Q107.467-43.937 107.458-43.919Q107.449-43.901 107.449-43.897Q107.133-43.502 106.663-43.280Q106.192-43.058 105.696-43.058Q105.287-43.058 104.905-43.168L104.905-41.449Q105.362-41.906 106.074-41.906Q106.584-41.906 106.983-41.625Q107.383-41.344 107.605-40.889Q107.827-40.434 107.827-39.929Q107.827-39.379 107.548-38.920Q107.269-38.461 106.803-38.195Q106.337-37.929 105.793-37.929Q105.353-37.929 104.969-38.156Q104.584-38.382 104.356-38.762Q104.127-39.142 104.127-39.586Q104.127-39.779 104.259-39.911Q104.391-40.043 104.588-40.043Q104.720-40.043 104.824-39.984Q104.927-39.924 104.986-39.821Q105.045-39.718 105.045-39.586Q105.045-39.388 104.918-39.256Q104.791-39.125 104.588-39.125Q104.527-39.125 104.496-39.133M108.974-35.868L108.891-35.868Q108.803-35.868 108.803-35.965Q108.803-36.004 108.838-36.040Q109.673-36.813 110.033-37.947Q110.394-39.081 110.394-40.377Q110.394-40.997 110.315-41.588Q110.235-42.179 110.055-42.737Q109.875-43.295 109.574-43.803Q109.273-44.310 108.838-44.706Q108.803-44.741 108.803-44.776Q108.803-44.877 108.891-44.877L108.974-44.877Q108.992-44.877 109.027-44.859Q109.532-44.477 109.906-43.963Q110.279-43.449 110.517-42.871Q110.754-42.293 110.870-41.665Q110.987-41.036 110.987-40.377Q110.987-39.718 110.870-39.087Q110.754-38.457 110.514-37.872Q110.275-37.288 109.904-36.778Q109.532-36.268 109.027-35.886Q108.992-35.868 108.974-35.868",[2846],[2830,14481,14482,14485],{"fill":2952,"stroke":2953,"style":2954},[2835,14483],{"d":14484},"M66.702 23.047h19.917V3.13H66.702Z",[2830,14486,14488],{"transform":14487},"translate(37.521 54.115)",[2835,14489],{"d":14352,"fill":2832,"stroke":2832,"className":14490,"style":2847},[2846],[2830,14492,14493,14496],{"fill":2952,"stroke":2953,"style":2954},[2835,14494],{"d":14495},"M112.226 23.047h19.917V3.13h-19.917Z",[2830,14497,14499],{"transform":14498},"translate(83.046 54.115)",[2835,14500],{"d":14436,"fill":2832,"stroke":2832,"className":14501,"style":2847},[2846],[2830,14503,14504],{"fill":2953,"stroke":2953},[2830,14505,14506,14513,14519,14525,14531,14537,14543,14549,14555,14561,14567,14573,14579,14585,14591,14597,14603],{"fill":2953,"stroke":2837,"fontFamily":4023,"fontSize":4024},[2830,14507,14509],{"transform":14508},"translate(-110.712 83.09)",[2835,14510],{"d":14511,"fill":2953,"stroke":2953,"className":14512,"style":4032},"M37.065-39.881Q37.065-40.361 37.298-40.777Q37.530-41.193 37.940-41.443Q38.350-41.693 38.827-41.693Q39.557-41.693 39.956-41.252Q40.354-40.811 40.354-40.080Q40.354-39.975 40.261-39.951L37.811-39.951L37.811-39.881Q37.811-39.471 37.932-39.115Q38.054-38.760 38.325-38.543Q38.597-38.326 39.026-38.326Q39.389-38.326 39.686-38.555Q39.983-38.783 40.085-39.135Q40.093-39.182 40.179-39.197L40.261-39.197Q40.354-39.170 40.354-39.088Q40.354-39.080 40.347-39.049Q40.284-38.822 40.145-38.639Q40.007-38.455 39.815-38.322Q39.624-38.190 39.405-38.119Q39.186-38.049 38.948-38.049Q38.577-38.049 38.239-38.186Q37.901-38.322 37.634-38.574Q37.366-38.826 37.216-39.166Q37.065-39.506 37.065-39.881M37.819-40.190L39.780-40.190Q39.780-40.494 39.679-40.785Q39.577-41.076 39.360-41.258Q39.143-41.440 38.827-41.440Q38.526-41.440 38.296-41.252Q38.065-41.065 37.942-40.773Q37.819-40.482 37.819-40.190M40.940-38.959Q40.940-39.443 41.343-39.738Q41.745-40.033 42.296-40.152Q42.847-40.272 43.339-40.272L43.339-40.561Q43.339-40.787 43.223-40.994Q43.108-41.201 42.911-41.320Q42.714-41.440 42.483-41.440Q42.057-41.440 41.772-41.334Q41.843-41.307 41.889-41.252Q41.936-41.197 41.962-41.127Q41.987-41.057 41.987-40.982Q41.987-40.877 41.936-40.785Q41.886-40.693 41.794-40.643Q41.702-40.592 41.597-40.592Q41.491-40.592 41.399-40.643Q41.307-40.693 41.257-40.785Q41.206-40.877 41.206-40.982Q41.206-41.400 41.595-41.547Q41.983-41.693 42.483-41.693Q42.815-41.693 43.169-41.563Q43.522-41.432 43.751-41.178Q43.979-40.924 43.979-40.576L43.979-38.775Q43.979-38.643 44.052-38.533Q44.124-38.424 44.253-38.424Q44.378-38.424 44.446-38.529Q44.514-38.635 44.514-38.775L44.514-39.287L44.796-39.287L44.796-38.775Q44.796-38.572 44.679-38.414Q44.561-38.256 44.380-38.172Q44.198-38.088 43.995-38.088Q43.764-38.088 43.612-38.260Q43.460-38.432 43.429-38.662Q43.268-38.381 42.960-38.215Q42.651-38.049 42.300-38.049Q41.788-38.049 41.364-38.272Q40.940-38.494 40.940-38.959M41.628-38.959Q41.628-38.674 41.854-38.488Q42.081-38.303 42.374-38.303Q42.620-38.303 42.845-38.420Q43.069-38.537 43.204-38.740Q43.339-38.943 43.339-39.197L43.339-40.029Q43.073-40.029 42.788-39.975Q42.503-39.920 42.231-39.791Q41.960-39.662 41.794-39.455Q41.628-39.248 41.628-38.959M45.132-39.854Q45.132-40.350 45.382-40.775Q45.632-41.201 46.052-41.447Q46.472-41.693 46.972-41.693Q47.511-41.693 47.901-41.568Q48.292-41.443 48.292-41.029Q48.292-40.924 48.241-40.832Q48.190-40.740 48.098-40.690Q48.007-40.639 47.897-40.639Q47.792-40.639 47.700-40.690Q47.608-40.740 47.557-40.832Q47.507-40.924 47.507-41.029Q47.507-41.252 47.675-41.357Q47.452-41.416 46.979-41.416Q46.682-41.416 46.468-41.277Q46.253-41.139 46.122-40.908Q45.991-40.678 45.932-40.408Q45.874-40.139 45.874-39.854Q45.874-39.459 46.007-39.109Q46.139-38.760 46.411-38.543Q46.682-38.326 47.081-38.326Q47.456-38.326 47.731-38.543Q48.007-38.760 48.108-39.119Q48.124-39.182 48.186-39.182L48.292-39.182Q48.327-39.182 48.352-39.154Q48.378-39.127 48.378-39.088L48.378-39.065Q48.245-38.584 47.860-38.316Q47.475-38.049 46.972-38.049Q46.608-38.049 46.274-38.186Q45.940-38.322 45.681-38.572Q45.421-38.822 45.276-39.158Q45.132-39.494 45.132-39.854",[2846],[2830,14514,14515],{"transform":14508},[2835,14516],{"d":14517,"fill":2953,"stroke":2953,"className":14518,"style":4032},"M50.564-38.127L48.709-38.127L48.709-38.424Q48.982-38.424 49.150-38.471Q49.318-38.518 49.318-38.686L49.318-42.846Q49.318-43.061 49.255-43.156Q49.193-43.252 49.074-43.273Q48.955-43.295 48.709-43.295L48.709-43.592L49.931-43.678L49.931-40.975Q50.056-41.186 50.244-41.336Q50.431-41.486 50.658-41.570Q50.884-41.654 51.130-41.654Q52.298-41.654 52.298-40.576L52.298-38.686Q52.298-38.518 52.468-38.471Q52.638-38.424 52.908-38.424L52.908-38.127L51.052-38.127L51.052-38.424Q51.326-38.424 51.494-38.471Q51.662-38.518 51.662-38.686L51.662-40.561Q51.662-40.943 51.541-41.172Q51.419-41.400 51.068-41.400Q50.755-41.400 50.501-41.238Q50.248-41.076 50.101-40.807Q49.955-40.537 49.955-40.240L49.955-38.686Q49.955-38.518 50.125-38.471Q50.294-38.424 50.564-38.424",[2846],[2830,14520,14521],{"transform":14508},[2835,14522],{"d":14523,"fill":2953,"stroke":2953,"className":14524,"style":4032},"M56.815-39.088L56.815-41.279L56.112-41.279L56.112-41.533Q56.468-41.533 56.710-41.766Q56.952-41.998 57.063-42.346Q57.175-42.693 57.175-43.049L57.456-43.049L57.456-41.576L58.632-41.576L58.632-41.279L57.456-41.279L57.456-39.104Q57.456-38.783 57.575-38.555Q57.694-38.326 57.975-38.326Q58.155-38.326 58.272-38.449Q58.389-38.572 58.442-38.752Q58.495-38.932 58.495-39.104L58.495-39.576L58.776-39.576L58.776-39.088Q58.776-38.834 58.671-38.594Q58.565-38.354 58.368-38.201Q58.171-38.049 57.913-38.049Q57.597-38.049 57.345-38.172Q57.093-38.295 56.954-38.529Q56.815-38.764 56.815-39.088M59.495-39.881Q59.495-40.361 59.727-40.777Q59.960-41.193 60.370-41.443Q60.780-41.693 61.257-41.693Q61.987-41.693 62.386-41.252Q62.784-40.811 62.784-40.080Q62.784-39.975 62.690-39.951L60.241-39.951L60.241-39.881Q60.241-39.471 60.362-39.115Q60.483-38.760 60.755-38.543Q61.026-38.326 61.456-38.326Q61.819-38.326 62.116-38.555Q62.413-38.783 62.514-39.135Q62.522-39.182 62.608-39.197L62.690-39.197Q62.784-39.170 62.784-39.088Q62.784-39.080 62.776-39.049Q62.714-38.822 62.575-38.639Q62.436-38.455 62.245-38.322Q62.054-38.190 61.835-38.119Q61.616-38.049 61.378-38.049Q61.007-38.049 60.669-38.186Q60.331-38.322 60.063-38.574Q59.796-38.826 59.645-39.166Q59.495-39.506 59.495-39.881M60.249-40.190L62.210-40.190Q62.210-40.494 62.108-40.785Q62.007-41.076 61.790-41.258Q61.573-41.440 61.257-41.440Q60.956-41.440 60.725-41.252Q60.495-41.065 60.372-40.773Q60.249-40.482 60.249-40.190M65.280-38.127L63.300-38.127L63.300-38.424Q63.569-38.424 63.737-38.469Q63.905-38.514 63.905-38.686L63.905-40.822Q63.905-41.037 63.843-41.133Q63.780-41.229 63.663-41.250Q63.546-41.272 63.300-41.272L63.300-41.568L64.468-41.654L64.468-40.869Q64.546-41.080 64.698-41.266Q64.850-41.451 65.050-41.553Q65.249-41.654 65.475-41.654Q65.722-41.654 65.913-41.510Q66.104-41.365 66.104-41.135Q66.104-40.979 65.999-40.869Q65.893-40.760 65.737-40.760Q65.581-40.760 65.472-40.869Q65.362-40.979 65.362-41.135Q65.362-41.295 65.468-41.400Q65.143-41.400 64.929-41.172Q64.714-40.943 64.618-40.604Q64.522-40.264 64.522-39.959L64.522-38.686Q64.522-38.518 64.749-38.471Q64.975-38.424 65.280-38.424L65.280-38.127M68.514-38.127L66.659-38.127L66.659-38.424Q66.932-38.424 67.100-38.471Q67.268-38.518 67.268-38.686L67.268-40.822Q67.268-41.037 67.206-41.133Q67.143-41.229 67.024-41.250Q66.905-41.272 66.659-41.272L66.659-41.568L67.850-41.654L67.850-40.920Q67.964-41.135 68.157-41.303Q68.350-41.471 68.589-41.563Q68.827-41.654 69.081-41.654Q70.042-41.654 70.218-40.943Q70.401-41.272 70.729-41.463Q71.057-41.654 71.436-41.654Q72.612-41.654 72.612-40.576L72.612-38.686Q72.612-38.518 72.780-38.471Q72.948-38.424 73.218-38.424L73.218-38.127L71.362-38.127L71.362-38.424Q71.636-38.424 71.804-38.469Q71.972-38.514 71.972-38.686L71.972-40.561Q71.972-40.947 71.847-41.174Q71.722-41.400 71.370-41.400Q71.065-41.400 70.809-41.238Q70.554-41.076 70.405-40.807Q70.257-40.537 70.257-40.240L70.257-38.686Q70.257-38.518 70.427-38.471Q70.597-38.424 70.866-38.424L70.866-38.127L69.011-38.127L69.011-38.424Q69.284-38.424 69.452-38.471Q69.620-38.518 69.620-38.686L69.620-40.561Q69.620-40.947 69.495-41.174Q69.370-41.400 69.018-41.400Q68.714-41.400 68.458-41.238Q68.202-41.076 68.054-40.807Q67.905-40.537 67.905-40.240L67.905-38.686Q67.905-38.518 68.075-38.471Q68.245-38.424 68.514-38.424",[2846],[2830,14526,14527],{"transform":14508},[2835,14528],{"d":14529,"fill":2953,"stroke":2953,"className":14530,"style":4032},"M78.363-38.127L76.585-38.127L76.585-38.424Q76.859-38.424 77.027-38.471Q77.195-38.518 77.195-38.686L77.195-40.822Q77.195-41.037 77.138-41.133Q77.081-41.229 76.968-41.250Q76.855-41.272 76.609-41.272L76.609-41.568L77.808-41.654L77.808-38.686Q77.808-38.518 77.954-38.471Q78.101-38.424 78.363-38.424L78.363-38.127M76.921-43.049Q76.921-43.240 77.056-43.371Q77.191-43.502 77.386-43.502Q77.507-43.502 77.611-43.440Q77.714-43.377 77.777-43.273Q77.839-43.170 77.839-43.049Q77.839-42.854 77.708-42.719Q77.578-42.584 77.386-42.584Q77.187-42.584 77.054-42.717Q76.921-42.850 76.921-43.049M78.906-38.135L78.906-39.357Q78.906-39.385 78.937-39.416Q78.968-39.447 78.992-39.447L79.097-39.447Q79.167-39.447 79.183-39.385Q79.245-39.065 79.384-38.824Q79.523-38.584 79.755-38.443Q79.988-38.303 80.296-38.303Q80.535-38.303 80.744-38.363Q80.953-38.424 81.089-38.572Q81.226-38.721 81.226-38.967Q81.226-39.221 81.015-39.387Q80.804-39.553 80.535-39.607L79.913-39.721Q79.507-39.799 79.206-40.055Q78.906-40.311 78.906-40.686Q78.906-41.053 79.107-41.275Q79.308-41.498 79.632-41.596Q79.956-41.693 80.296-41.693Q80.761-41.693 81.058-41.486L81.281-41.670Q81.304-41.693 81.335-41.693L81.386-41.693Q81.417-41.693 81.445-41.666Q81.472-41.639 81.472-41.607L81.472-40.623Q81.472-40.592 81.447-40.563Q81.421-40.533 81.386-40.533L81.281-40.533Q81.245-40.533 81.218-40.561Q81.191-40.588 81.191-40.623Q81.191-41.022 80.939-41.242Q80.687-41.463 80.288-41.463Q79.933-41.463 79.650-41.340Q79.367-41.217 79.367-40.912Q79.367-40.693 79.568-40.561Q79.769-40.428 80.015-40.385L80.640-40.272Q81.070-40.182 81.378-39.885Q81.687-39.588 81.687-39.174Q81.687-38.604 81.288-38.326Q80.890-38.049 80.296-38.049Q79.745-38.049 79.394-38.385L79.097-38.072Q79.074-38.049 79.038-38.049L78.992-38.049Q78.968-38.049 78.937-38.080Q78.906-38.111 78.906-38.135",[2846],[2830,14532,14533],{"transform":14508},[2835,14534],{"d":14535,"fill":2953,"stroke":2953,"className":14536,"style":4032},"M85.148-38.959Q85.148-39.443 85.550-39.738Q85.953-40.033 86.503-40.152Q87.054-40.272 87.546-40.272L87.546-40.561Q87.546-40.787 87.431-40.994Q87.316-41.201 87.119-41.320Q86.921-41.440 86.691-41.440Q86.265-41.440 85.980-41.334Q86.050-41.307 86.097-41.252Q86.144-41.197 86.169-41.127Q86.195-41.057 86.195-40.982Q86.195-40.877 86.144-40.785Q86.093-40.693 86.001-40.643Q85.910-40.592 85.804-40.592Q85.699-40.592 85.607-40.643Q85.515-40.693 85.464-40.785Q85.414-40.877 85.414-40.982Q85.414-41.400 85.802-41.547Q86.191-41.693 86.691-41.693Q87.023-41.693 87.376-41.563Q87.730-41.432 87.958-41.178Q88.187-40.924 88.187-40.576L88.187-38.775Q88.187-38.643 88.259-38.533Q88.332-38.424 88.460-38.424Q88.585-38.424 88.654-38.529Q88.722-38.635 88.722-38.775L88.722-39.287L89.003-39.287L89.003-38.775Q89.003-38.572 88.886-38.414Q88.769-38.256 88.587-38.172Q88.406-38.088 88.203-38.088Q87.972-38.088 87.820-38.260Q87.667-38.432 87.636-38.662Q87.476-38.381 87.167-38.215Q86.859-38.049 86.507-38.049Q85.996-38.049 85.572-38.272Q85.148-38.494 85.148-38.959M85.835-38.959Q85.835-38.674 86.062-38.488Q86.289-38.303 86.582-38.303Q86.828-38.303 87.052-38.420Q87.277-38.537 87.412-38.740Q87.546-38.943 87.546-39.197L87.546-40.029Q87.281-40.029 86.996-39.975Q86.710-39.920 86.439-39.791Q86.167-39.662 86.001-39.455Q85.835-39.248 85.835-38.959",[2846],[2830,14538,14539],{"transform":14508},[2835,14540],{"d":14541,"fill":2953,"stroke":2953,"className":14542,"style":4032},"M92.177-38.135L92.177-39.357Q92.177-39.385 92.209-39.416Q92.240-39.447 92.263-39.447L92.369-39.447Q92.439-39.447 92.455-39.385Q92.517-39.065 92.656-38.824Q92.794-38.584 93.027-38.443Q93.259-38.303 93.568-38.303Q93.806-38.303 94.015-38.363Q94.224-38.424 94.361-38.572Q94.498-38.721 94.498-38.967Q94.498-39.221 94.287-39.387Q94.076-39.553 93.806-39.607L93.185-39.721Q92.779-39.799 92.478-40.055Q92.177-40.311 92.177-40.686Q92.177-41.053 92.378-41.275Q92.580-41.498 92.904-41.596Q93.228-41.693 93.568-41.693Q94.033-41.693 94.330-41.486L94.552-41.670Q94.576-41.693 94.607-41.693L94.658-41.693Q94.689-41.693 94.716-41.666Q94.744-41.639 94.744-41.607L94.744-40.623Q94.744-40.592 94.718-40.563Q94.693-40.533 94.658-40.533L94.552-40.533Q94.517-40.533 94.490-40.561Q94.462-40.588 94.462-40.623Q94.462-41.022 94.210-41.242Q93.959-41.463 93.560-41.463Q93.205-41.463 92.921-41.340Q92.638-41.217 92.638-40.912Q92.638-40.693 92.839-40.561Q93.041-40.428 93.287-40.385L93.912-40.272Q94.341-40.182 94.650-39.885Q94.959-39.588 94.959-39.174Q94.959-38.604 94.560-38.326Q94.162-38.049 93.568-38.049Q93.017-38.049 92.666-38.385L92.369-38.072Q92.345-38.049 92.310-38.049L92.263-38.049Q92.240-38.049 92.209-38.080Q92.177-38.111 92.177-38.135M95.486-39.881Q95.486-40.361 95.718-40.777Q95.951-41.193 96.361-41.443Q96.771-41.693 97.248-41.693Q97.978-41.693 98.376-41.252Q98.775-40.811 98.775-40.080Q98.775-39.975 98.681-39.951L96.232-39.951L96.232-39.881Q96.232-39.471 96.353-39.115Q96.474-38.760 96.746-38.543Q97.017-38.326 97.447-38.326Q97.810-38.326 98.107-38.555Q98.404-38.783 98.505-39.135Q98.513-39.182 98.599-39.197L98.681-39.197Q98.775-39.170 98.775-39.088Q98.775-39.080 98.767-39.049Q98.705-38.822 98.566-38.639Q98.427-38.455 98.236-38.322Q98.044-38.190 97.826-38.119Q97.607-38.049 97.369-38.049Q96.998-38.049 96.660-38.186Q96.322-38.322 96.054-38.574Q95.787-38.826 95.636-39.166Q95.486-39.506 95.486-39.881M96.240-40.190L98.201-40.190Q98.201-40.494 98.099-40.785Q97.998-41.076 97.781-41.258Q97.564-41.440 97.248-41.440Q96.947-41.440 96.716-41.252Q96.486-41.065 96.363-40.773Q96.240-40.482 96.240-40.190M101.177-38.127L99.345-38.127L99.345-38.424Q99.619-38.424 99.787-38.471Q99.955-38.518 99.955-38.686L99.955-42.846Q99.955-43.061 99.892-43.156Q99.830-43.252 99.710-43.273Q99.591-43.295 99.345-43.295L99.345-43.592L100.568-43.678L100.568-38.686Q100.568-38.518 100.736-38.471Q100.904-38.424 101.177-38.424L101.177-38.127M101.623-39.881Q101.623-40.361 101.855-40.777Q102.087-41.193 102.498-41.443Q102.908-41.693 103.384-41.693Q104.115-41.693 104.513-41.252Q104.912-40.811 104.912-40.080Q104.912-39.975 104.818-39.951L102.369-39.951L102.369-39.881Q102.369-39.471 102.490-39.115Q102.611-38.760 102.882-38.543Q103.154-38.326 103.584-38.326Q103.947-38.326 104.244-38.555Q104.541-38.783 104.642-39.135Q104.650-39.182 104.736-39.197L104.818-39.197Q104.912-39.170 104.912-39.088Q104.912-39.080 104.904-39.049Q104.841-38.822 104.703-38.639Q104.564-38.455 104.373-38.322Q104.181-38.190 103.962-38.119Q103.744-38.049 103.505-38.049Q103.134-38.049 102.796-38.186Q102.459-38.322 102.191-38.574Q101.923-38.826 101.773-39.166Q101.623-39.506 101.623-39.881M102.376-40.190L104.337-40.190Q104.337-40.494 104.236-40.785Q104.134-41.076 103.917-41.258Q103.701-41.440 103.384-41.440Q103.084-41.440 102.853-41.252Q102.623-41.065 102.500-40.773Q102.376-40.482 102.376-40.190M105.443-39.854Q105.443-40.350 105.693-40.775Q105.943-41.201 106.363-41.447Q106.783-41.693 107.283-41.693Q107.822-41.693 108.212-41.568Q108.603-41.443 108.603-41.029Q108.603-40.924 108.552-40.832Q108.501-40.740 108.410-40.690Q108.318-40.639 108.209-40.639Q108.103-40.639 108.011-40.690Q107.919-40.740 107.869-40.832Q107.818-40.924 107.818-41.029Q107.818-41.252 107.986-41.357Q107.763-41.416 107.291-41.416Q106.994-41.416 106.779-41.277Q106.564-41.139 106.433-40.908Q106.302-40.678 106.244-40.408Q106.185-40.139 106.185-39.854Q106.185-39.459 106.318-39.109Q106.451-38.760 106.722-38.543Q106.994-38.326 107.392-38.326Q107.767-38.326 108.042-38.543Q108.318-38.760 108.419-39.119Q108.435-39.182 108.498-39.182L108.603-39.182Q108.638-39.182 108.664-39.154Q108.689-39.127 108.689-39.088L108.689-39.065Q108.556-38.584 108.171-38.316Q107.787-38.049 107.283-38.049Q106.919-38.049 106.585-38.186Q106.251-38.322 105.992-38.572Q105.732-38.822 105.587-39.158Q105.443-39.494 105.443-39.854M109.802-39.088L109.802-41.279L109.099-41.279L109.099-41.533Q109.455-41.533 109.697-41.766Q109.939-41.998 110.050-42.346Q110.162-42.693 110.162-43.049L110.443-43.049L110.443-41.576L111.619-41.576L111.619-41.279L110.443-41.279L110.443-39.104Q110.443-38.783 110.562-38.555Q110.681-38.326 110.962-38.326Q111.142-38.326 111.259-38.449Q111.376-38.572 111.429-38.752Q111.482-38.932 111.482-39.104L111.482-39.576L111.763-39.576L111.763-39.088Q111.763-38.834 111.658-38.594Q111.552-38.354 111.355-38.201Q111.158-38.049 110.900-38.049Q110.584-38.049 110.332-38.172Q110.080-38.295 109.941-38.529Q109.802-38.764 109.802-39.088M112.482-39.822Q112.482-40.326 112.738-40.758Q112.994-41.190 113.429-41.441Q113.865-41.693 114.365-41.693Q114.751-41.693 115.093-41.549Q115.435-41.404 115.697-41.143Q115.959-40.881 116.101-40.545Q116.244-40.209 116.244-39.822Q116.244-39.330 115.980-38.920Q115.716-38.510 115.287-38.279Q114.857-38.049 114.365-38.049Q113.873-38.049 113.439-38.281Q113.005-38.514 112.744-38.922Q112.482-39.330 112.482-39.822M114.365-38.326Q114.822-38.326 115.074-38.549Q115.326-38.772 115.414-39.123Q115.501-39.475 115.501-39.920Q115.501-40.350 115.408-40.688Q115.314-41.025 115.060-41.232Q114.806-41.440 114.365-41.440Q113.716-41.440 113.472-41.023Q113.228-40.607 113.228-39.920Q113.228-39.475 113.316-39.123Q113.404-38.772 113.656-38.549Q113.908-38.326 114.365-38.326M118.736-38.127L116.755-38.127L116.755-38.424Q117.025-38.424 117.193-38.469Q117.361-38.514 117.361-38.686L117.361-40.822Q117.361-41.037 117.298-41.133Q117.236-41.229 117.119-41.250Q117.001-41.272 116.755-41.272L116.755-41.568L117.923-41.654L117.923-40.869Q118.001-41.080 118.154-41.266Q118.306-41.451 118.505-41.553Q118.705-41.654 118.931-41.654Q119.177-41.654 119.369-41.510Q119.560-41.365 119.560-41.135Q119.560-40.979 119.455-40.869Q119.349-40.760 119.193-40.760Q119.037-40.760 118.927-40.869Q118.818-40.979 118.818-41.135Q118.818-41.295 118.923-41.400Q118.599-41.400 118.384-41.172Q118.169-40.943 118.074-40.604Q117.978-40.264 117.978-39.959L117.978-38.686Q117.978-38.518 118.205-38.471Q118.431-38.424 118.736-38.424L118.736-38.127M120.521-38.592Q120.521-38.775 120.658-38.912Q120.794-39.049 120.986-39.049Q121.177-39.049 121.310-38.916Q121.443-38.783 121.443-38.592Q121.443-38.393 121.310-38.260Q121.177-38.127 120.986-38.127Q120.794-38.127 120.658-38.264Q120.521-38.400 120.521-38.592M120.521-41.119Q120.521-41.303 120.658-41.440Q120.794-41.576 120.986-41.576Q121.177-41.576 121.310-41.443Q121.443-41.311 121.443-41.119Q121.443-40.920 121.310-40.787Q121.177-40.654 120.986-40.654Q120.794-40.654 120.658-40.791Q120.521-40.928 120.521-41.119",[2846],[2830,14544,14545],{"transform":14508},[2835,14546],{"d":14547,"fill":2953,"stroke":2953,"className":14548,"style":4032},"M129.544-38.127L126.751-38.127L126.751-38.424Q127.813-38.424 127.813-38.686L127.813-42.854Q127.384-42.639 126.704-42.639L126.704-42.936Q127.723-42.936 128.239-43.447L128.384-43.447Q128.458-43.428 128.477-43.350L128.477-38.686Q128.477-38.424 129.544-38.424",[2846],[2830,14550,14551],{"transform":14508},[2835,14552],{"d":14553,"fill":2953,"stroke":2953,"className":14554,"style":4032},"M133.272-39.822Q133.272-40.326 133.528-40.758Q133.784-41.190 134.220-41.441Q134.655-41.693 135.155-41.693Q135.542-41.693 135.884-41.549Q136.225-41.404 136.487-41.143Q136.749-40.881 136.891-40.545Q137.034-40.209 137.034-39.822Q137.034-39.330 136.770-38.920Q136.507-38.510 136.077-38.279Q135.647-38.049 135.155-38.049Q134.663-38.049 134.229-38.281Q133.796-38.514 133.534-38.922Q133.272-39.330 133.272-39.822M135.155-38.326Q135.612-38.326 135.864-38.549Q136.116-38.772 136.204-39.123Q136.292-39.475 136.292-39.920Q136.292-40.350 136.198-40.688Q136.104-41.025 135.850-41.232Q135.596-41.440 135.155-41.440Q134.507-41.440 134.263-41.023Q134.018-40.607 134.018-39.920Q134.018-39.475 134.106-39.123Q134.194-38.772 134.446-38.549Q134.698-38.326 135.155-38.326M139.448-38.127L137.593-38.127L137.593-38.424Q137.866-38.424 138.034-38.471Q138.202-38.518 138.202-38.686L138.202-40.822Q138.202-41.037 138.139-41.133Q138.077-41.229 137.958-41.250Q137.839-41.272 137.593-41.272L137.593-41.568L138.784-41.654L138.784-40.920Q138.897-41.135 139.091-41.303Q139.284-41.471 139.522-41.563Q139.761-41.654 140.014-41.654Q141.182-41.654 141.182-40.576L141.182-38.686Q141.182-38.518 141.352-38.471Q141.522-38.424 141.792-38.424L141.792-38.127L139.936-38.127L139.936-38.424Q140.210-38.424 140.378-38.471Q140.546-38.518 140.546-38.686L140.546-40.561Q140.546-40.943 140.425-41.172Q140.304-41.400 139.952-41.400Q139.639-41.400 139.386-41.238Q139.132-41.076 138.985-40.807Q138.839-40.537 138.839-40.240L138.839-38.686Q138.839-38.518 139.009-38.471Q139.179-38.424 139.448-38.424",[2846],[2830,14556,14557],{"transform":14508},[2835,14558],{"d":14559,"fill":2953,"stroke":2953,"className":14560,"style":4032},"M146.937-38.127L145.159-38.127L145.159-38.424Q145.433-38.424 145.601-38.471Q145.769-38.518 145.769-38.686L145.769-40.822Q145.769-41.037 145.712-41.133Q145.655-41.229 145.542-41.250Q145.429-41.272 145.183-41.272L145.183-41.568L146.382-41.654L146.382-38.686Q146.382-38.518 146.528-38.471Q146.675-38.424 146.937-38.424L146.937-38.127M145.495-43.049Q145.495-43.240 145.630-43.371Q145.765-43.502 145.960-43.502Q146.081-43.502 146.185-43.440Q146.288-43.377 146.351-43.273Q146.413-43.170 146.413-43.049Q146.413-42.854 146.282-42.719Q146.151-42.584 145.960-42.584Q145.761-42.584 145.628-42.717Q145.495-42.850 145.495-43.049M148.062-39.088L148.062-41.279L147.359-41.279L147.359-41.533Q147.714-41.533 147.956-41.766Q148.198-41.998 148.310-42.346Q148.421-42.693 148.421-43.049L148.702-43.049L148.702-41.576L149.878-41.576L149.878-41.279L148.702-41.279L148.702-39.104Q148.702-38.783 148.821-38.555Q148.941-38.326 149.222-38.326Q149.401-38.326 149.519-38.449Q149.636-38.572 149.689-38.752Q149.741-38.932 149.741-39.104L149.741-39.576L150.023-39.576L150.023-39.088Q150.023-38.834 149.917-38.594Q149.812-38.354 149.614-38.201Q149.417-38.049 149.159-38.049Q148.843-38.049 148.591-38.172Q148.339-38.295 148.200-38.529Q148.062-38.764 148.062-39.088M150.784-38.135L150.784-39.357Q150.784-39.385 150.816-39.416Q150.847-39.447 150.870-39.447L150.976-39.447Q151.046-39.447 151.062-39.385Q151.124-39.065 151.263-38.824Q151.401-38.584 151.634-38.443Q151.866-38.303 152.175-38.303Q152.413-38.303 152.622-38.363Q152.831-38.424 152.968-38.572Q153.105-38.721 153.105-38.967Q153.105-39.221 152.894-39.387Q152.683-39.553 152.413-39.607L151.792-39.721Q151.386-39.799 151.085-40.055Q150.784-40.311 150.784-40.686Q150.784-41.053 150.985-41.275Q151.187-41.498 151.511-41.596Q151.835-41.693 152.175-41.693Q152.640-41.693 152.937-41.486L153.159-41.670Q153.183-41.693 153.214-41.693L153.265-41.693Q153.296-41.693 153.323-41.666Q153.351-41.639 153.351-41.607L153.351-40.623Q153.351-40.592 153.325-40.563Q153.300-40.533 153.265-40.533L153.159-40.533Q153.124-40.533 153.097-40.561Q153.069-40.588 153.069-40.623Q153.069-41.022 152.818-41.242Q152.566-41.463 152.167-41.463Q151.812-41.463 151.528-41.340Q151.245-41.217 151.245-40.912Q151.245-40.693 151.446-40.561Q151.648-40.428 151.894-40.385L152.519-40.272Q152.948-40.182 153.257-39.885Q153.566-39.588 153.566-39.174Q153.566-38.604 153.167-38.326Q152.769-38.049 152.175-38.049Q151.624-38.049 151.273-38.385L150.976-38.072Q150.952-38.049 150.917-38.049L150.870-38.049Q150.847-38.049 150.816-38.080Q150.784-38.111 150.784-38.135",[2846],[2830,14562,14563],{"transform":14508},[2835,14564],{"d":14565,"fill":2953,"stroke":2953,"className":14566,"style":4032},"M156.930-39.822Q156.930-40.326 157.186-40.758Q157.442-41.190 157.878-41.441Q158.313-41.693 158.813-41.693Q159.200-41.693 159.542-41.549Q159.883-41.404 160.145-41.143Q160.407-40.881 160.549-40.545Q160.692-40.209 160.692-39.822Q160.692-39.330 160.428-38.920Q160.165-38.510 159.735-38.279Q159.305-38.049 158.813-38.049Q158.321-38.049 157.887-38.281Q157.454-38.514 157.192-38.922Q156.930-39.330 156.930-39.822M158.813-38.326Q159.270-38.326 159.522-38.549Q159.774-38.772 159.862-39.123Q159.950-39.475 159.950-39.920Q159.950-40.350 159.856-40.688Q159.762-41.025 159.508-41.232Q159.255-41.440 158.813-41.440Q158.165-41.440 157.921-41.023Q157.676-40.607 157.676-39.920Q157.676-39.475 157.764-39.123Q157.852-38.772 158.104-38.549Q158.356-38.326 158.813-38.326",[2846],[2830,14568,14569],{"transform":14508},[2835,14570],{"d":14571,"fill":2953,"stroke":2953,"className":14572,"style":4032},"M162.530-38.158L161.460-41.014Q161.393-41.193 161.263-41.236Q161.132-41.279 160.874-41.279L160.874-41.576L162.554-41.576L162.554-41.279Q162.104-41.279 162.104-41.080Q162.108-41.065 162.110-41.047Q162.112-41.029 162.112-41.014L162.905-38.920L163.616-40.830Q163.581-40.924 163.581-40.969Q163.581-41.014 163.546-41.014Q163.479-41.193 163.349-41.236Q163.218-41.279 162.964-41.279L162.964-41.576L164.554-41.576L164.554-41.279Q164.104-41.279 164.104-41.080Q164.108-41.061 164.110-41.043Q164.112-41.025 164.112-41.014L164.944-38.799L165.698-40.799Q165.722-40.857 165.722-40.928Q165.722-41.088 165.585-41.184Q165.448-41.279 165.280-41.279L165.280-41.576L166.667-41.576L166.667-41.279Q166.433-41.279 166.255-41.152Q166.077-41.025 165.995-40.799L165.011-38.158Q164.956-38.049 164.843-38.049L164.784-38.049Q164.671-38.049 164.628-38.158L163.768-40.432L162.913-38.158Q162.874-38.049 162.753-38.049L162.698-38.049Q162.585-38.049 162.530-38.158M169.011-38.127L167.155-38.127L167.155-38.424Q167.429-38.424 167.597-38.471Q167.765-38.518 167.765-38.686L167.765-40.822Q167.765-41.037 167.702-41.133Q167.640-41.229 167.520-41.250Q167.401-41.272 167.155-41.272L167.155-41.568L168.347-41.654L168.347-40.920Q168.460-41.135 168.653-41.303Q168.847-41.471 169.085-41.563Q169.323-41.654 169.577-41.654Q170.745-41.654 170.745-40.576L170.745-38.686Q170.745-38.518 170.915-38.471Q171.085-38.424 171.354-38.424L171.354-38.127L169.499-38.127L169.499-38.424Q169.772-38.424 169.940-38.471Q170.108-38.518 170.108-38.686L170.108-40.561Q170.108-40.943 169.987-41.172Q169.866-41.400 169.515-41.400Q169.202-41.400 168.948-41.238Q168.694-41.076 168.548-40.807Q168.401-40.537 168.401-40.240L168.401-38.686Q168.401-38.518 168.571-38.471Q168.741-38.424 169.011-38.424",[2846],[2830,14574,14575],{"transform":14508},[2835,14576],{"d":14577,"fill":2953,"stroke":2953,"className":14578,"style":4032},"M176.569-38.127L174.714-38.127L174.714-38.424Q174.987-38.424 175.155-38.471Q175.323-38.518 175.323-38.686L175.323-40.822Q175.323-41.037 175.260-41.133Q175.198-41.229 175.079-41.250Q174.960-41.272 174.714-41.272L174.714-41.568L175.905-41.654L175.905-40.920Q176.018-41.135 176.212-41.303Q176.405-41.471 176.643-41.563Q176.881-41.654 177.135-41.654Q178.096-41.654 178.272-40.943Q178.456-41.272 178.784-41.463Q179.112-41.654 179.491-41.654Q180.667-41.654 180.667-40.576L180.667-38.686Q180.667-38.518 180.835-38.471Q181.003-38.424 181.272-38.424L181.272-38.127L179.417-38.127L179.417-38.424Q179.690-38.424 179.858-38.469Q180.026-38.514 180.026-38.686L180.026-40.561Q180.026-40.947 179.901-41.174Q179.776-41.400 179.424-41.400Q179.120-41.400 178.864-41.238Q178.608-41.076 178.460-40.807Q178.311-40.537 178.311-40.240L178.311-38.686Q178.311-38.518 178.481-38.471Q178.651-38.424 178.921-38.424L178.921-38.127L177.065-38.127L177.065-38.424Q177.339-38.424 177.506-38.471Q177.674-38.518 177.674-38.686L177.674-40.561Q177.674-40.947 177.549-41.174Q177.424-41.400 177.073-41.400Q176.768-41.400 176.512-41.238Q176.256-41.076 176.108-40.807Q175.960-40.537 175.960-40.240L175.960-38.686Q175.960-38.518 176.130-38.471Q176.299-38.424 176.569-38.424L176.569-38.127M181.717-39.822Q181.717-40.326 181.973-40.758Q182.229-41.190 182.665-41.441Q183.100-41.693 183.600-41.693Q183.987-41.693 184.329-41.549Q184.671-41.404 184.932-41.143Q185.194-40.881 185.337-40.545Q185.479-40.209 185.479-39.822Q185.479-39.330 185.215-38.920Q184.952-38.510 184.522-38.279Q184.092-38.049 183.600-38.049Q183.108-38.049 182.674-38.281Q182.241-38.514 181.979-38.922Q181.717-39.330 181.717-39.822M183.600-38.326Q184.057-38.326 184.309-38.549Q184.561-38.772 184.649-39.123Q184.737-39.475 184.737-39.920Q184.737-40.350 184.643-40.688Q184.549-41.025 184.296-41.232Q184.042-41.440 183.600-41.440Q182.952-41.440 182.708-41.023Q182.464-40.607 182.464-39.920Q182.464-39.475 182.551-39.123Q182.639-38.772 182.891-38.549Q183.143-38.326 183.600-38.326",[2846],[2830,14580,14581],{"transform":14508},[2835,14582],{"d":14583,"fill":2953,"stroke":2953,"className":14584,"style":4032},"M188.026-38.049Q187.545-38.049 187.137-38.293Q186.729-38.537 186.491-38.951Q186.252-39.365 186.252-39.854Q186.252-40.346 186.510-40.762Q186.768-41.178 187.200-41.416Q187.631-41.654 188.123-41.654Q188.744-41.654 189.194-41.217L189.194-42.846Q189.194-43.061 189.131-43.156Q189.069-43.252 188.951-43.273Q188.834-43.295 188.588-43.295L188.588-43.592L189.811-43.678L189.811-38.869Q189.811-38.658 189.873-38.563Q189.936-38.467 190.053-38.445Q190.170-38.424 190.420-38.424L190.420-38.127L189.170-38.049L189.170-38.533Q188.705-38.049 188.026-38.049M188.092-38.303Q188.432-38.303 188.725-38.494Q189.018-38.686 189.170-38.982L189.170-40.815Q189.022-41.088 188.760-41.244Q188.498-41.400 188.186-41.400Q187.561-41.400 187.278-40.953Q186.994-40.506 186.994-39.846Q186.994-39.201 187.246-38.752Q187.498-38.303 188.092-38.303M191.612-39.080L191.612-40.822Q191.612-41.037 191.549-41.133Q191.487-41.229 191.367-41.250Q191.248-41.272 191.002-41.272L191.002-41.568L192.248-41.654L192.248-39.104L192.248-39.080Q192.248-38.768 192.303-38.606Q192.358-38.443 192.508-38.373Q192.659-38.303 192.979-38.303Q193.409-38.303 193.682-38.641Q193.955-38.979 193.955-39.424L193.955-40.822Q193.955-41.037 193.893-41.133Q193.830-41.229 193.711-41.250Q193.592-41.272 193.346-41.272L193.346-41.568L194.592-41.654L194.592-38.869Q194.592-38.658 194.655-38.563Q194.717-38.467 194.836-38.445Q194.955-38.424 195.201-38.424L195.201-38.127L193.979-38.049L193.979-38.670Q193.811-38.381 193.530-38.215Q193.248-38.049 192.928-38.049Q191.612-38.049 191.612-39.080M197.561-38.127L195.729-38.127L195.729-38.424Q196.002-38.424 196.170-38.471Q196.338-38.518 196.338-38.686L196.338-42.846Q196.338-43.061 196.276-43.156Q196.213-43.252 196.094-43.273Q195.975-43.295 195.729-43.295L195.729-43.592L196.951-43.678L196.951-38.686Q196.951-38.518 197.119-38.471Q197.287-38.424 197.561-38.424L197.561-38.127M198.690-39.080L198.690-40.822Q198.690-41.037 198.627-41.133Q198.565-41.229 198.446-41.250Q198.326-41.272 198.080-41.272L198.080-41.568L199.326-41.654L199.326-39.104L199.326-39.080Q199.326-38.768 199.381-38.606Q199.436-38.443 199.586-38.373Q199.737-38.303 200.057-38.303Q200.487-38.303 200.760-38.641Q201.034-38.979 201.034-39.424L201.034-40.822Q201.034-41.037 200.971-41.133Q200.909-41.229 200.789-41.250Q200.670-41.272 200.424-41.272L200.424-41.568L201.670-41.654L201.670-38.869Q201.670-38.658 201.733-38.563Q201.795-38.467 201.914-38.445Q202.034-38.424 202.280-38.424L202.280-38.127L201.057-38.049L201.057-38.670Q200.889-38.381 200.608-38.215Q200.326-38.049 200.006-38.049Q198.690-38.049 198.690-39.080M202.768-38.135L202.768-39.357Q202.768-39.385 202.799-39.416Q202.830-39.447 202.854-39.447L202.959-39.447Q203.030-39.447 203.045-39.385Q203.108-39.065 203.246-38.824Q203.385-38.584 203.617-38.443Q203.850-38.303 204.159-38.303Q204.397-38.303 204.606-38.363Q204.815-38.424 204.951-38.572Q205.088-38.721 205.088-38.967Q205.088-39.221 204.877-39.387Q204.666-39.553 204.397-39.607L203.776-39.721Q203.369-39.799 203.069-40.055Q202.768-40.311 202.768-40.686Q202.768-41.053 202.969-41.275Q203.170-41.498 203.494-41.596Q203.819-41.693 204.159-41.693Q204.623-41.693 204.920-41.486L205.143-41.670Q205.166-41.693 205.198-41.693L205.248-41.693Q205.280-41.693 205.307-41.666Q205.334-41.639 205.334-41.607L205.334-40.623Q205.334-40.592 205.309-40.563Q205.284-40.533 205.248-40.533L205.143-40.533Q205.108-40.533 205.080-40.561Q205.053-40.588 205.053-40.623Q205.053-41.022 204.801-41.242Q204.549-41.463 204.151-41.463Q203.795-41.463 203.512-41.340Q203.229-41.217 203.229-40.912Q203.229-40.693 203.430-40.561Q203.631-40.428 203.877-40.385L204.502-40.272Q204.932-40.182 205.241-39.885Q205.549-39.588 205.549-39.174Q205.549-38.604 205.151-38.326Q204.752-38.049 204.159-38.049Q203.608-38.049 203.256-38.385L202.959-38.072Q202.936-38.049 202.901-38.049L202.854-38.049Q202.830-38.049 202.799-38.080Q202.768-38.111 202.768-38.135M206.662-36.721Q206.662-36.744 206.694-36.791Q206.987-37.053 207.153-37.420Q207.319-37.787 207.319-38.174L207.319-38.232Q207.190-38.127 207.022-38.127Q206.830-38.127 206.694-38.260Q206.557-38.393 206.557-38.592Q206.557-38.783 206.694-38.916Q206.830-39.049 207.022-39.049Q207.323-39.049 207.448-38.779Q207.573-38.510 207.573-38.174Q207.573-37.725 207.391-37.311Q207.209-36.897 206.869-36.600Q206.846-36.576 206.807-36.576Q206.760-36.576 206.711-36.621Q206.662-36.666 206.662-36.721",[2846],[2830,14586,14587],{"transform":14508},[2835,14588],{"d":14589,"fill":2953,"stroke":2953,"className":14590,"style":4032},"M213.167-37.959Q212.464-37.959 212.064-38.359Q211.663-38.760 211.519-39.369Q211.374-39.979 211.374-40.678Q211.374-41.201 211.444-41.664Q211.515-42.127 211.708-42.539Q211.901-42.951 212.259-43.199Q212.616-43.447 213.167-43.447Q213.718-43.447 214.075-43.199Q214.433-42.951 214.624-42.541Q214.816-42.131 214.886-41.662Q214.956-41.193 214.956-40.678Q214.956-39.979 214.814-39.371Q214.671-38.764 214.271-38.361Q213.870-37.959 213.167-37.959M213.167-38.217Q213.640-38.217 213.872-38.652Q214.105-39.088 214.159-39.627Q214.214-40.166 214.214-40.807Q214.214-41.803 214.030-42.496Q213.847-43.190 213.167-43.190Q212.800-43.190 212.579-42.951Q212.358-42.713 212.263-42.356Q212.167-41.998 212.142-41.627Q212.116-41.256 212.116-40.807Q212.116-40.166 212.171-39.627Q212.226-39.088 212.458-38.652Q212.691-38.217 213.167-38.217",[2846],[2830,14592,14593],{"transform":14508},[2835,14594],{"d":14595,"fill":2953,"stroke":2953,"className":14596,"style":4032},"M218.367-39.822Q218.367-40.326 218.623-40.758Q218.879-41.190 219.315-41.441Q219.750-41.693 220.250-41.693Q220.637-41.693 220.979-41.549Q221.320-41.404 221.582-41.143Q221.844-40.881 221.986-40.545Q222.129-40.209 222.129-39.822Q222.129-39.330 221.865-38.920Q221.602-38.510 221.172-38.279Q220.742-38.049 220.250-38.049Q219.758-38.049 219.324-38.281Q218.891-38.514 218.629-38.922Q218.367-39.330 218.367-39.822M220.250-38.326Q220.707-38.326 220.959-38.549Q221.211-38.772 221.299-39.123Q221.387-39.475 221.387-39.920Q221.387-40.350 221.293-40.688Q221.199-41.025 220.945-41.232Q220.691-41.440 220.250-41.440Q219.602-41.440 219.358-41.023Q219.113-40.607 219.113-39.920Q219.113-39.475 219.201-39.123Q219.289-38.772 219.541-38.549Q219.793-38.326 220.250-38.326M224.543-38.127L222.688-38.127L222.688-38.424Q222.961-38.424 223.129-38.471Q223.297-38.518 223.297-38.686L223.297-40.822Q223.297-41.037 223.234-41.133Q223.172-41.229 223.053-41.250Q222.934-41.272 222.688-41.272L222.688-41.568L223.879-41.654L223.879-40.920Q223.992-41.135 224.186-41.303Q224.379-41.471 224.617-41.563Q224.856-41.654 225.109-41.654Q226.277-41.654 226.277-40.576L226.277-38.686Q226.277-38.518 226.447-38.471Q226.617-38.424 226.887-38.424L226.887-38.127L225.031-38.127L225.031-38.424Q225.305-38.424 225.473-38.471Q225.641-38.518 225.641-38.686L225.641-40.561Q225.641-40.943 225.520-41.172Q225.399-41.400 225.047-41.400Q224.734-41.400 224.481-41.238Q224.227-41.076 224.080-40.807Q223.934-40.537 223.934-40.240L223.934-38.686Q223.934-38.518 224.104-38.471Q224.274-38.424 224.543-38.424",[2846],[2830,14598,14599],{"transform":14508},[2835,14600],{"d":14601,"fill":2953,"stroke":2953,"className":14602,"style":4032},"M230.798-39.088L230.798-41.279L230.095-41.279L230.095-41.533Q230.451-41.533 230.693-41.766Q230.935-41.998 231.046-42.346Q231.158-42.693 231.158-43.049L231.439-43.049L231.439-41.576L232.615-41.576L232.615-41.279L231.439-41.279L231.439-39.104Q231.439-38.783 231.558-38.555Q231.677-38.326 231.958-38.326Q232.138-38.326 232.255-38.449Q232.373-38.572 232.425-38.752Q232.478-38.932 232.478-39.104L232.478-39.576L232.759-39.576L232.759-39.088Q232.759-38.834 232.654-38.594Q232.548-38.354 232.351-38.201Q232.154-38.049 231.896-38.049Q231.580-38.049 231.328-38.172Q231.076-38.295 230.937-38.529Q230.798-38.764 230.798-39.088M235.408-38.127L233.552-38.127L233.552-38.424Q233.826-38.424 233.994-38.471Q234.162-38.518 234.162-38.686L234.162-42.846Q234.162-43.061 234.099-43.156Q234.037-43.252 233.917-43.273Q233.798-43.295 233.552-43.295L233.552-43.592L234.775-43.678L234.775-40.975Q234.900-41.186 235.087-41.336Q235.275-41.486 235.501-41.570Q235.728-41.654 235.974-41.654Q237.142-41.654 237.142-40.576L237.142-38.686Q237.142-38.518 237.312-38.471Q237.482-38.424 237.751-38.424L237.751-38.127L235.896-38.127L235.896-38.424Q236.169-38.424 236.337-38.471Q236.505-38.518 236.505-38.686L236.505-40.561Q236.505-40.943 236.384-41.172Q236.263-41.400 235.912-41.400Q235.599-41.400 235.345-41.238Q235.091-41.076 234.945-40.807Q234.798-40.537 234.798-40.240L234.798-38.686Q234.798-38.518 234.968-38.471Q235.138-38.424 235.408-38.424L235.408-38.127M238.197-39.881Q238.197-40.361 238.429-40.777Q238.662-41.193 239.072-41.443Q239.482-41.693 239.958-41.693Q240.689-41.693 241.087-41.252Q241.486-40.811 241.486-40.080Q241.486-39.975 241.392-39.951L238.943-39.951L238.943-39.881Q238.943-39.471 239.064-39.115Q239.185-38.760 239.456-38.543Q239.728-38.326 240.158-38.326Q240.521-38.326 240.818-38.555Q241.115-38.783 241.216-39.135Q241.224-39.182 241.310-39.197L241.392-39.197Q241.486-39.170 241.486-39.088Q241.486-39.080 241.478-39.049Q241.415-38.822 241.277-38.639Q241.138-38.455 240.947-38.322Q240.755-38.190 240.537-38.119Q240.318-38.049 240.080-38.049Q239.708-38.049 239.371-38.186Q239.033-38.322 238.765-38.574Q238.498-38.826 238.347-39.166Q238.197-39.506 238.197-39.881M238.951-40.190L240.912-40.190Q240.912-40.494 240.810-40.785Q240.708-41.076 240.492-41.258Q240.275-41.440 239.958-41.440Q239.658-41.440 239.427-41.252Q239.197-41.065 239.074-40.773Q238.951-40.482 238.951-40.190",[2846],[2830,14604,14605],{"transform":14508},[2835,14606],{"d":14607,"fill":2953,"stroke":2953,"className":14608,"style":4032},"M244.812-39.822Q244.812-40.326 245.068-40.758Q245.324-41.190 245.760-41.441Q246.195-41.693 246.695-41.693Q247.082-41.693 247.424-41.549Q247.765-41.404 248.027-41.143Q248.289-40.881 248.431-40.545Q248.574-40.209 248.574-39.822Q248.574-39.330 248.310-38.920Q248.047-38.510 247.617-38.279Q247.187-38.049 246.695-38.049Q246.203-38.049 245.769-38.281Q245.336-38.514 245.074-38.922Q244.812-39.330 244.812-39.822M246.695-38.326Q247.152-38.326 247.404-38.549Q247.656-38.772 247.744-39.123Q247.832-39.475 247.832-39.920Q247.832-40.350 247.738-40.688Q247.644-41.025 247.390-41.232Q247.137-41.440 246.695-41.440Q246.047-41.440 245.803-41.023Q245.558-40.607 245.558-39.920Q245.558-39.475 245.646-39.123Q245.734-38.772 245.986-38.549Q246.238-38.326 246.695-38.326M249.683-39.088L249.683-41.279L248.980-41.279L248.980-41.533Q249.336-41.533 249.578-41.766Q249.820-41.998 249.931-42.346Q250.043-42.693 250.043-43.049L250.324-43.049L250.324-41.576L251.500-41.576L251.500-41.279L250.324-41.279L250.324-39.104Q250.324-38.783 250.443-38.555Q250.562-38.326 250.844-38.326Q251.023-38.326 251.140-38.449Q251.258-38.572 251.310-38.752Q251.363-38.932 251.363-39.104L251.363-39.576L251.644-39.576L251.644-39.088Q251.644-38.834 251.539-38.594Q251.433-38.354 251.236-38.201Q251.039-38.049 250.781-38.049Q250.465-38.049 250.213-38.172Q249.961-38.295 249.822-38.529Q249.683-38.764 249.683-39.088M254.293-38.127L252.437-38.127L252.437-38.424Q252.711-38.424 252.879-38.471Q253.047-38.518 253.047-38.686L253.047-42.846Q253.047-43.061 252.984-43.156Q252.922-43.252 252.803-43.273Q252.683-43.295 252.437-43.295L252.437-43.592L253.660-43.678L253.660-40.975Q253.785-41.186 253.972-41.336Q254.160-41.486 254.387-41.570Q254.613-41.654 254.859-41.654Q256.027-41.654 256.027-40.576L256.027-38.686Q256.027-38.518 256.197-38.471Q256.367-38.424 256.637-38.424L256.637-38.127L254.781-38.127L254.781-38.424Q255.054-38.424 255.222-38.471Q255.390-38.518 255.390-38.686L255.390-40.561Q255.390-40.943 255.269-41.172Q255.148-41.400 254.797-41.400Q254.484-41.400 254.230-41.238Q253.976-41.076 253.830-40.807Q253.683-40.537 253.683-40.240L253.683-38.686Q253.683-38.518 253.853-38.471Q254.023-38.424 254.293-38.424L254.293-38.127M257.082-39.881Q257.082-40.361 257.314-40.777Q257.547-41.193 257.957-41.443Q258.367-41.693 258.844-41.693Q259.574-41.693 259.972-41.252Q260.371-40.811 260.371-40.080Q260.371-39.975 260.277-39.951L257.828-39.951L257.828-39.881Q257.828-39.471 257.949-39.115Q258.070-38.760 258.342-38.543Q258.613-38.326 259.043-38.326Q259.406-38.326 259.703-38.555Q260-38.783 260.101-39.135Q260.109-39.182 260.195-39.197L260.277-39.197Q260.371-39.170 260.371-39.088Q260.371-39.080 260.363-39.049Q260.301-38.822 260.162-38.639Q260.023-38.455 259.832-38.322Q259.640-38.190 259.422-38.119Q259.203-38.049 258.965-38.049Q258.594-38.049 258.256-38.186Q257.918-38.322 257.650-38.574Q257.383-38.826 257.232-39.166Q257.082-39.506 257.082-39.881M257.836-40.190L259.797-40.190Q259.797-40.494 259.695-40.785Q259.594-41.076 259.377-41.258Q259.160-41.440 258.844-41.440Q258.543-41.440 258.312-41.252Q258.082-41.065 257.959-40.773Q257.836-40.482 257.836-40.190M262.867-38.127L260.887-38.127L260.887-38.424Q261.156-38.424 261.324-38.469Q261.492-38.514 261.492-38.686L261.492-40.822Q261.492-41.037 261.429-41.133Q261.367-41.229 261.250-41.250Q261.133-41.272 260.887-41.272L260.887-41.568L262.054-41.654L262.054-40.869Q262.133-41.080 262.285-41.266Q262.437-41.451 262.637-41.553Q262.836-41.654 263.062-41.654Q263.308-41.654 263.500-41.510Q263.691-41.365 263.691-41.135Q263.691-40.979 263.586-40.869Q263.480-40.760 263.324-40.760Q263.168-40.760 263.058-40.869Q262.949-40.979 262.949-41.135Q262.949-41.295 263.054-41.400Q262.730-41.400 262.515-41.172Q262.301-40.943 262.205-40.604Q262.109-40.264 262.109-39.959L262.109-38.686Q262.109-38.518 262.336-38.471Q262.562-38.424 262.867-38.424",[2846],[3003,14610,14612,14613,14762,14763,14778,14779,1099],{"className":14611},[3006],"CRT as a sum of selectors: each ",[394,14614,14616],{"className":14615},[397],[394,14617,14619],{"className":14618,"ariaHidden":402},[401],[394,14620,14622,14625,14665,14705],{"className":14621},[406],[394,14623],{"className":14624,"style":12807},[410],[394,14626,14628,14631],{"className":14627},[415],[394,14629,384],{"className":14630},[415,419],[394,14632,14634],{"className":14633},[423],[394,14635,14637,14657],{"className":14636},[427,913],[394,14638,14640,14654],{"className":14639},[431],[394,14641,14643],{"className":14642,"style":5200},[435],[394,14644,14645,14648],{"style":5016},[394,14646],{"className":14647,"style":443},[442],[394,14649,14651],{"className":14650},[447,448,449,450],[394,14652,5212],{"className":14653},[415,419,450],[394,14655,983],{"className":14656},[982],[394,14658,14660],{"className":14659},[431],[394,14661,14663],{"className":14662,"style":5035},[435],[394,14664],{},[394,14666,14668,14671],{"className":14667},[415],[394,14669,12165],{"className":14670,"style":7567},[415,419],[394,14672,14674],{"className":14673},[423],[394,14675,14677,14697],{"className":14676},[427,913],[394,14678,14680,14694],{"className":14679},[431],[394,14681,14683],{"className":14682,"style":5200},[435],[394,14684,14685,14688],{"style":12305},[394,14686],{"className":14687,"style":443},[442],[394,14689,14691],{"className":14690},[447,448,449,450],[394,14692,5212],{"className":14693},[415,419,450],[394,14695,983],{"className":14696},[982],[394,14698,14700],{"className":14699},[431],[394,14701,14703],{"className":14702,"style":5035},[435],[394,14704],{},[394,14706,14708,14711],{"className":14707},[415],[394,14709,12165],{"className":14710,"style":7567},[415,419],[394,14712,14714],{"className":14713},[423],[394,14715,14717,14754],{"className":14716},[427,913],[394,14718,14720,14751],{"className":14719},[431],[394,14721,14723,14734],{"className":14722,"style":12826},[435],[394,14724,14725,14728],{"style":12829},[394,14726],{"className":14727,"style":443},[442],[394,14729,14731],{"className":14730},[447,448,449,450],[394,14732,5212],{"className":14733},[415,419,450],[394,14735,14736,14739],{"style":12841},[394,14737],{"className":14738,"style":443},[442],[394,14740,14742],{"className":14741},[447,448,449,450],[394,14743,14745,14748],{"className":14744},[415,450],[394,14746,457],{"className":14747},[415,450],[394,14749,461],{"className":14750},[415,450],[394,14752,983],{"className":14753},[982],[394,14755,14757],{"className":14756},[431],[394,14758,14760],{"className":14759,"style":12866},[435],[394,14761],{}," hits its own residue and is ",[394,14764,14766],{"className":14765},[397],[394,14767,14769],{"className":14768,"ariaHidden":402},[401],[394,14770,14772,14775],{"className":14771},[406],[394,14773],{"className":14774,"style":639},[410],[394,14776,780],{"className":14777},[415]," modulo the other, so the terms add to ",[394,14780,14782],{"className":14781},[397],[394,14783,14785,14803,14818],{"className":14784,"ariaHidden":402},[401],[394,14786,14788,14791,14794,14797,14800],{"className":14787},[406],[394,14789],{"className":14790,"style":11721},[410],[394,14792,4243],{"className":14793},[415,419],[394,14795],{"className":14796,"style":466},[465],[394,14798,471],{"className":14799},[470],[394,14801],{"className":14802,"style":466},[465],[394,14804,14806,14809,14812,14815],{"className":14805},[406],[394,14807],{"className":14808,"style":639},[410],[394,14810,4024],{"className":14811},[415],[394,14813],{"className":14814},[465,524],[394,14816],{"className":14817,"style":528},[465],[394,14819,14821,14824,14827,14836,14839,14842],{"className":14820},[406],[394,14822],{"className":14823,"style":535},[410],[394,14825,540],{"className":14826},[539],[394,14828,14830],{"className":14829},[415],[394,14831,14833],{"className":14832},[415],[394,14834,551],{"className":14835},[415,550],[394,14837],{"className":14838,"style":555},[465],[394,14840,10660],{"className":14841},[415],[394,14843,563],{"className":14844},[562],[7692,14846,14848],{"className":7694,"code":14847,"language":7696,"meta":376,"style":376},"caption: $\\textsc{CRT}(a[\\,], m[\\,])$ — combine congruences with pairwise-coprime moduli\n$M \\gets \\prod_i m_i$\n$x \\gets 0$\nfor $i \\gets 1$ to $n$ do\n  $M_i \\gets M \u002F m_i$\n  $y_i \\gets \\textsc{Mod-Inverse}(M_i \\bmod m_i,\\; m_i)$\n  $x \\gets (x + a_i \\cdot M_i \\cdot y_i) \\bmod M$\nreturn $x$\n",[7698,14849,14850,14855,14860,14865,14870,14875,14880,14885],{"__ignoreMap":376},[394,14851,14852],{"class":7702,"line":6},[394,14853,14854],{},"caption: $\\textsc{CRT}(a[\\,], m[\\,])$ — combine congruences with pairwise-coprime moduli\n",[394,14856,14857],{"class":7702,"line":18},[394,14858,14859],{},"$M \\gets \\prod_i m_i$\n",[394,14861,14862],{"class":7702,"line":24},[394,14863,14864],{},"$x \\gets 0$\n",[394,14866,14867],{"class":7702,"line":73},[394,14868,14869],{},"for $i \\gets 1$ to $n$ do\n",[394,14871,14872],{"class":7702,"line":102},[394,14873,14874],{},"  $M_i \\gets M \u002F m_i$\n",[394,14876,14877],{"class":7702,"line":108},[394,14878,14879],{},"  $y_i \\gets \\textsc{Mod-Inverse}(M_i \\bmod m_i,\\; m_i)$\n",[394,14881,14882],{"class":7702,"line":116},[394,14883,14884],{},"  $x \\gets (x + a_i \\cdot M_i \\cdot y_i) \\bmod M$\n",[394,14886,14887],{"class":7702,"line":196},[394,14888,14889],{},"return $x$\n",[381,14891,14892,14893,14908],{},"CRT is what lets us compute modulo a large composite ",[394,14894,14896],{"className":14895},[397],[394,14897,14899],{"className":14898,"ariaHidden":402},[401],[394,14900,14902,14905],{"className":14901},[406],[394,14903],{"className":14904,"style":871},[410],[394,14906,12165],{"className":14907,"style":7567},[415,419]," by working independently in\neach prime-power factor and reassembling, the structural twin of how stars-and-bars\nor inclusion–exclusion decompose a hard count into manageable pieces.",[578,14910,14912],{"id":14911},"takeaways","Takeaways",[14914,14915,14916,15181,15511,15758,15972,15977],"ul",{},[14917,14918,14919,14922,14923,14941,14942,15014,15015,15018,15019,15161,15162,15180],"li",{},[389,14920,14921],{},"Permutations"," count orderings (",[394,14924,14926],{"className":14925},[397],[394,14927,14929],{"className":14928,"ariaHidden":402},[401],[394,14930,14932,14935,14938],{"className":14931},[406],[394,14933],{"className":14934,"style":660},[410],[394,14936,605],{"className":14937},[415,419],[394,14939,667],{"className":14940},[562],", or ",[394,14943,14945],{"className":14944},[397],[394,14946,14948,14972,15002],{"className":14947,"ariaHidden":402},[401],[394,14949,14951,14954,14957,14960,14963,14966,14969],{"className":14950},[406],[394,14952],{"className":14953,"style":871},[410],[394,14955,605],{"className":14956},[415,419],[394,14958,879],{"className":14959,"style":878},[415,419],[394,14961,822],{"className":14962,"style":821},[415,419],[394,14964],{"className":14965,"style":466},[465],[394,14967,674],{"className":14968},[470],[394,14970],{"className":14971,"style":466},[465],[394,14973,14975,14978,14981,14984,14987,14990,14993,14996,14999],{"className":14974},[406],[394,14976],{"className":14977,"style":535},[410],[394,14979,605],{"className":14980},[415,419],[394,14982,667],{"className":14983},[562],[394,14985,11685],{"className":14986},[415],[394,14988,540],{"className":14989},[539],[394,14991,605],{"className":14992},[415,419],[394,14994],{"className":14995,"style":626},[465],[394,14997,457],{"className":14998},[516],[394,15000],{"className":15001,"style":626},[465],[394,15003,15005,15008,15011],{"className":15004},[406],[394,15006],{"className":15007,"style":535},[410],[394,15009,822],{"className":15010,"style":821},[415,419],[394,15012,952],{"className":15013},[562],"); ",[389,15016,15017],{},"combinations","\ncount subsets, ",[394,15020,15022],{"className":15021},[397],[394,15023,15025,15106,15148],{"className":15024,"ariaHidden":402},[401],[394,15026,15028,15031,15097,15100,15103],{"className":15027},[406],[394,15029],{"className":15030,"style":1497},[410],[394,15032,15034,15040,15091],{"className":15033},[415],[394,15035,15037],{"className":15036,"style":1226},[539,1225],[394,15038,540],{"className":15039},[1230,1507],[394,15041,15043],{"className":15042},[909],[394,15044,15046,15083],{"className":15045},[427,913],[394,15047,15049,15080],{"className":15048},[431],[394,15050,15052,15066],{"className":15051,"style":1520},[435],[394,15053,15054,15057],{"style":1523},[394,15055],{"className":15056,"style":443},[442],[394,15058,15060],{"className":15059},[447,448,449,450],[394,15061,15063],{"className":15062},[415,450],[394,15064,822],{"className":15065,"style":821},[415,419,450],[394,15067,15068,15071],{"style":1538},[394,15069],{"className":15070,"style":443},[442],[394,15072,15074],{"className":15073},[447,448,449,450],[394,15075,15077],{"className":15076},[415,450],[394,15078,605],{"className":15079},[415,419,450],[394,15081,983],{"className":15082},[982],[394,15084,15086],{"className":15085},[431],[394,15087,15089],{"className":15088,"style":1560},[435],[394,15090],{},[394,15092,15094],{"className":15093,"style":1226},[562,1225],[394,15095,563],{"className":15096},[1230,1507],[394,15098],{"className":15099,"style":466},[465],[394,15101,674],{"className":15102},[470],[394,15104],{"className":15105,"style":466},[465],[394,15107,15109,15112,15115,15118,15121,15124,15127,15130,15133,15136,15139,15142,15145],{"className":15108},[406],[394,15110],{"className":15111,"style":535},[410],[394,15113,605],{"className":15114},[415,419],[394,15116,667],{"className":15117},[562],[394,15119,11685],{"className":15120},[415],[394,15122,540],{"className":15123},[539],[394,15125,822],{"className":15126,"style":821},[415,419],[394,15128,667],{"className":15129},[562],[394,15131],{"className":15132,"style":734},[465],[394,15134,540],{"className":15135},[539],[394,15137,605],{"className":15138},[415,419],[394,15140],{"className":15141,"style":626},[465],[394,15143,457],{"className":15144},[516],[394,15146],{"className":15147,"style":626},[465],[394,15149,15151,15154,15157],{"className":15150},[406],[394,15152],{"className":15153,"style":535},[410],[394,15155,822],{"className":15156,"style":821},[415,419],[394,15158,15160],{"className":15159},[562],")!)",", dividing out the ",[394,15163,15165],{"className":15164},[397],[394,15166,15168],{"className":15167,"ariaHidden":402},[401],[394,15169,15171,15174,15177],{"className":15170},[406],[394,15172],{"className":15173,"style":660},[410],[394,15175,822],{"className":15176,"style":821},[415,419],[394,15178,667],{"className":15179},[562]," orderings.",[14917,15182,15183,392,15185,15443,15444,15459,15460,15510],{},[389,15184,2017],{},[394,15186,15188],{"className":15187},[397],[394,15189,15191,15272,15365],{"className":15190,"ariaHidden":402},[401],[394,15192,15194,15197,15263,15266,15269],{"className":15193},[406],[394,15195],{"className":15196,"style":1497},[410],[394,15198,15200,15206,15257],{"className":15199},[415],[394,15201,15203],{"className":15202,"style":1226},[539,1225],[394,15204,540],{"className":15205},[1230,1507],[394,15207,15209],{"className":15208},[909],[394,15210,15212,15249],{"className":15211},[427,913],[394,15213,15215,15246],{"className":15214},[431],[394,15216,15218,15232],{"className":15217,"style":1520},[435],[394,15219,15220,15223],{"style":1523},[394,15221],{"className":15222,"style":443},[442],[394,15224,15226],{"className":15225},[447,448,449,450],[394,15227,15229],{"className":15228},[415,450],[394,15230,2068],{"className":15231,"style":2067},[415,419,450],[394,15233,15234,15237],{"style":1538},[394,15235],{"className":15236,"style":443},[442],[394,15238,15240],{"className":15239},[447,448,449,450],[394,15241,15243],{"className":15242},[415,450],[394,15244,605],{"className":15245},[415,419,450],[394,15247,983],{"className":15248},[982],[394,15250,15252],{"className":15251},[431],[394,15253,15255],{"className":15254,"style":1560},[435],[394,15256],{},[394,15258,15260],{"className":15259,"style":1226},[562,1225],[394,15261,563],{"className":15262},[1230,1507],[394,15264],{"className":15265,"style":466},[465],[394,15267,674],{"className":15268},[470],[394,15270],{"className":15271,"style":466},[465],[394,15273,15275,15278,15356,15359,15362],{"className":15274},[406],[394,15276],{"className":15277,"style":2181},[410],[394,15279,15281,15287,15350],{"className":15280},[415],[394,15282,15284],{"className":15283,"style":1226},[539,1225],[394,15285,540],{"className":15286},[1230,1507],[394,15288,15290],{"className":15289},[909],[394,15291,15293,15342],{"className":15292},[427,913],[394,15294,15296,15339],{"className":15295},[431],[394,15297,15299,15319],{"className":15298,"style":2203},[435],[394,15300,15301,15304],{"style":1523},[394,15302],{"className":15303,"style":443},[442],[394,15305,15307],{"className":15306},[447,448,449,450],[394,15308,15310,15313,15316],{"className":15309},[415,450],[394,15311,2068],{"className":15312,"style":2067},[415,419,450],[394,15314,457],{"className":15315},[516,450],[394,15317,461],{"className":15318},[415,450],[394,15320,15321,15324],{"style":1538},[394,15322],{"className":15323,"style":443},[442],[394,15325,15327],{"className":15326},[447,448,449,450],[394,15328,15330,15333,15336],{"className":15329},[415,450],[394,15331,605],{"className":15332},[415,419,450],[394,15334,457],{"className":15335},[516,450],[394,15337,461],{"className":15338},[415,450],[394,15340,983],{"className":15341},[982],[394,15343,15345],{"className":15344},[431],[394,15346,15348],{"className":15347,"style":1776},[435],[394,15349],{},[394,15351,15353],{"className":15352,"style":1226},[562,1225],[394,15354,563],{"className":15355},[1230,1507],[394,15357],{"className":15358,"style":626},[465],[394,15360,1080],{"className":15361},[516],[394,15363],{"className":15364,"style":626},[465],[394,15366,15368,15371],{"className":15367},[406],[394,15369],{"className":15370,"style":2276},[410],[394,15372,15374,15380,15437],{"className":15373},[415],[394,15375,15377],{"className":15376,"style":1226},[539,1225],[394,15378,540],{"className":15379},[1230,1507],[394,15381,15383],{"className":15382},[909],[394,15384,15386,15429],{"className":15385},[427,913],[394,15387,15389,15426],{"className":15388},[431],[394,15390,15392,15406],{"className":15391,"style":2203},[435],[394,15393,15394,15397],{"style":1523},[394,15395],{"className":15396,"style":443},[442],[394,15398,15400],{"className":15399},[447,448,449,450],[394,15401,15403],{"className":15402},[415,450],[394,15404,2068],{"className":15405,"style":2067},[415,419,450],[394,15407,15408,15411],{"style":1538},[394,15409],{"className":15410,"style":443},[442],[394,15412,15414],{"className":15413},[447,448,449,450],[394,15415,15417,15420,15423],{"className":15416},[415,450],[394,15418,605],{"className":15419},[415,419,450],[394,15421,457],{"className":15422},[516,450],[394,15424,461],{"className":15425},[415,450],[394,15427,983],{"className":15428},[982],[394,15430,15432],{"className":15431},[431],[394,15433,15435],{"className":15434,"style":1560},[435],[394,15436],{},[394,15438,15440],{"className":15439,"style":1226},[562,1225],[394,15441,563],{"className":15442},[1230,1507]," (element ",[394,15445,15447],{"className":15446},[397],[394,15448,15450],{"className":15449,"ariaHidden":402},[401],[394,15451,15453,15456],{"className":15452},[406],[394,15454],{"className":15455,"style":601},[410],[394,15457,605],{"className":15458},[415,419]," in\nor out) builds the triangle additively in ",[394,15461,15463],{"className":15462},[397],[394,15464,15466],{"className":15465,"ariaHidden":402},[401],[394,15467,15469,15472,15475,15478,15507],{"className":15468},[406],[394,15470],{"className":15471,"style":3596},[410],[394,15473,3600],{"className":15474,"style":821},[415,419],[394,15476,540],{"className":15477},[539],[394,15479,15481,15484],{"className":15480},[415],[394,15482,605],{"className":15483},[415,419],[394,15485,15487],{"className":15486},[423],[394,15488,15490],{"className":15489},[427],[394,15491,15493],{"className":15492},[431],[394,15494,15496],{"className":15495,"style":411},[435],[394,15497,15498,15501],{"style":438},[394,15499],{"className":15500,"style":443},[442],[394,15502,15504],{"className":15503},[447,448,449,450],[394,15505,520],{"className":15506},[415,450],[394,15508,563],{"className":15509},[562]," with no division.",[14917,15512,15513,15516,15517,15660,15661,15757],{},[389,15514,15515],{},"Stars and bars",": the ordered non-negative solutions of ",[394,15518,15520],{"className":15519},[397],[394,15521,15523,15578,15596,15651],{"className":15522,"ariaHidden":402},[401],[394,15524,15526,15529,15569,15572,15575],{"className":15525},[406],[394,15527],{"className":15528,"style":4994},[410],[394,15530,15532,15535],{"className":15531},[415],[394,15533,4243],{"className":15534},[415,419],[394,15536,15538],{"className":15537},[423],[394,15539,15541,15561],{"className":15540},[427,913],[394,15542,15544,15558],{"className":15543},[431],[394,15545,15547],{"className":15546,"style":5013},[435],[394,15548,15549,15552],{"style":5016},[394,15550],{"className":15551,"style":443},[442],[394,15553,15555],{"className":15554},[447,448,449,450],[394,15556,461],{"className":15557},[415,450],[394,15559,983],{"className":15560},[982],[394,15562,15564],{"className":15563},[431],[394,15565,15567],{"className":15566,"style":5035},[435],[394,15568],{},[394,15570],{"className":15571,"style":626},[465],[394,15573,1080],{"className":15574},[516],[394,15576],{"className":15577,"style":626},[465],[394,15579,15581,15584,15587,15590,15593],{"className":15580},[406],[394,15582],{"className":15583,"style":619},[410],[394,15585,739],{"className":15586},[738],[394,15588],{"className":15589,"style":626},[465],[394,15591,1080],{"className":15592},[516],[394,15594],{"className":15595,"style":626},[465],[394,15597,15599,15602,15642,15645,15648],{"className":15598},[406],[394,15600],{"className":15601,"style":5126},[410],[394,15603,15605,15608],{"className":15604},[415],[394,15606,4243],{"className":15607},[415,419],[394,15609,15611],{"className":15610},[423],[394,15612,15614,15634],{"className":15613},[427,913],[394,15615,15617,15631],{"className":15616},[431],[394,15618,15620],{"className":15619,"style":5145},[435],[394,15621,15622,15625],{"style":5016},[394,15623],{"className":15624,"style":443},[442],[394,15626,15628],{"className":15627},[447,448,449,450],[394,15629,2068],{"className":15630,"style":2067},[415,419,450],[394,15632,983],{"className":15633},[982],[394,15635,15637],{"className":15636},[431],[394,15638,15640],{"className":15639,"style":5035},[435],[394,15641],{},[394,15643],{"className":15644,"style":466},[465],[394,15646,674],{"className":15647},[470],[394,15649],{"className":15650,"style":466},[465],[394,15652,15654,15657],{"className":15653},[406],[394,15655],{"className":15656,"style":601},[410],[394,15658,605],{"className":15659},[415,419]," number\n",[394,15662,15664],{"className":15663},[397],[394,15665,15667],{"className":15666,"ariaHidden":402},[401],[394,15668,15670,15673],{"className":15669},[406],[394,15671],{"className":15672,"style":5296},[410],[394,15674,15676,15682,15751],{"className":15675},[415],[394,15677,15679],{"className":15678,"style":1226},[539,1225],[394,15680,540],{"className":15681},[1230,1507],[394,15683,15685],{"className":15684},[909],[394,15686,15688,15743],{"className":15687},[427,913],[394,15689,15691,15740],{"className":15690},[431],[394,15692,15694,15714],{"className":15693,"style":5318},[435],[394,15695,15696,15699],{"style":1523},[394,15697],{"className":15698,"style":443},[442],[394,15700,15702],{"className":15701},[447,448,449,450],[394,15703,15705,15708,15711],{"className":15704},[415,450],[394,15706,2068],{"className":15707,"style":2067},[415,419,450],[394,15709,457],{"className":15710},[516,450],[394,15712,461],{"className":15713},[415,450],[394,15715,15716,15719],{"style":1538},[394,15717],{"className":15718,"style":443},[442],[394,15720,15722],{"className":15721},[447,448,449,450],[394,15723,15725,15728,15731,15734,15737],{"className":15724},[415,450],[394,15726,605],{"className":15727},[415,419,450],[394,15729,1080],{"className":15730},[516,450],[394,15732,2068],{"className":15733,"style":2067},[415,419,450],[394,15735,457],{"className":15736},[516,450],[394,15738,461],{"className":15739},[415,450],[394,15741,983],{"className":15742},[982],[394,15744,15746],{"className":15745},[431],[394,15747,15749],{"className":15748,"style":1776},[435],[394,15750],{},[394,15752,15754],{"className":15753,"style":1226},[562,1225],[394,15755,563],{"className":15756},[1230,1507],", via the bars-between-stars bijection.",[14917,15759,15760,15761,15869,15870,6249,15873,15876,15877,2700,15901,15925,15926,13110,15929,1099],{},"To compute ",[394,15762,15764],{"className":15763},[397],[394,15765,15767,15860],{"className":15766,"ariaHidden":402},[401],[394,15768,15770,15773,15839,15842,15845,15854,15857],{"className":15769},[406],[394,15771],{"className":15772,"style":1497},[410],[394,15774,15776,15782,15833],{"className":15775},[415],[394,15777,15779],{"className":15778,"style":1226},[539,1225],[394,15780,540],{"className":15781},[1230,1507],[394,15783,15785],{"className":15784},[909],[394,15786,15788,15825],{"className":15787},[427,913],[394,15789,15791,15822],{"className":15790},[431],[394,15792,15794,15808],{"className":15793,"style":1520},[435],[394,15795,15796,15799],{"style":1523},[394,15797],{"className":15798,"style":443},[442],[394,15800,15802],{"className":15801},[447,448,449,450],[394,15803,15805],{"className":15804},[415,450],[394,15806,2068],{"className":15807,"style":2067},[415,419,450],[394,15809,15810,15813],{"style":1538},[394,15811],{"className":15812,"style":443},[442],[394,15814,15816],{"className":15815},[447,448,449,450],[394,15817,15819],{"className":15818},[415,450],[394,15820,605],{"className":15821},[415,419,450],[394,15823,983],{"className":15824},[982],[394,15826,15828],{"className":15827},[431],[394,15829,15831],{"className":15830,"style":1560},[435],[394,15832],{},[394,15834,15836],{"className":15835,"style":1226},[562,1225],[394,15837,563],{"className":15838},[1230,1507],[394,15840],{"className":15841,"style":7042},[465],[394,15843],{"className":15844,"style":626},[465],[394,15846,15848],{"className":15847},[516],[394,15849,15851],{"className":15850},[415],[394,15852,551],{"className":15853},[415,550],[394,15855],{"className":15856,"style":7042},[465],[394,15858],{"className":15859,"style":626},[465],[394,15861,15863,15866],{"className":15862},[406],[394,15864],{"className":15865,"style":4597},[410],[394,15867,381],{"className":15868},[415,419],", precompute ",[389,15871,15872],{},"factorials",[389,15874,15875],{},"inverse\nfactorials"," (one Fermat inverse, then peel factors) in ",[394,15878,15880],{"className":15879},[397],[394,15881,15883],{"className":15882,"ariaHidden":402},[401],[394,15884,15886,15889,15892,15895,15898],{"className":15885},[406],[394,15887],{"className":15888,"style":535},[410],[394,15890,3600],{"className":15891,"style":821},[415,419],[394,15893,540],{"className":15894},[539],[394,15896,605],{"className":15897},[415,419],[394,15899,563],{"className":15900},[562],[394,15902,15904],{"className":15903},[397],[394,15905,15907],{"className":15906,"ariaHidden":402},[401],[394,15908,15910,15913,15916,15919,15922],{"className":15909},[406],[394,15911],{"className":15912,"style":535},[410],[394,15914,3600],{"className":15915,"style":821},[415,419],[394,15917,540],{"className":15918},[539],[394,15920,461],{"className":15921},[415],[394,15923,563],{"className":15924},[562]," per\nquery; use ",[389,15927,15928],{},"Lucas' theorem",[394,15930,15932],{"className":15931},[397],[394,15933,15935,15963],{"className":15934,"ariaHidden":402},[401],[394,15936,15938,15941,15944,15947,15950,15953,15956,15960],{"className":15937},[406],[394,15939],{"className":15940,"style":761},[410],[394,15942,605],{"className":15943},[415,419],[394,15945,769],{"className":15946},[768],[394,15948],{"className":15949,"style":734},[465],[394,15951,2068],{"className":15952,"style":2067},[415,419],[394,15954],{"className":15955,"style":466},[465],[394,15957,15959],{"className":15958},[470],">",[394,15961],{"className":15962,"style":466},[465],[394,15964,15966,15969],{"className":15965},[406],[394,15967],{"className":15968,"style":4597},[410],[394,15970,381],{"className":15971},[415,419],[14917,15973,15974,15976],{},[389,15975,9039],{}," counts unions by alternating add\u002Fsubtract over all\nintersections; the alternating signs make each element net-counted exactly once.",[14917,15978,15979,15980,15982,15983,16224],{},"The ",[389,15981,575],{}," uniquely solves a system of congruences with\ncoprime moduli via ",[394,15984,15986],{"className":15985},[397],[394,15987,15989,16007,16215],{"className":15988,"ariaHidden":402},[401],[394,15990,15992,15995,15998,16001,16004],{"className":15991},[406],[394,15993],{"className":15994,"style":601},[410],[394,15996,4243],{"className":15997},[415,419],[394,15999],{"className":16000,"style":466},[465],[394,16002,674],{"className":16003},[470],[394,16005],{"className":16006,"style":466},[465],[394,16008,16010,16014,16054,16057,16097,16137,16194,16197,16200,16209,16212],{"className":16009},[406],[394,16011],{"className":16012,"style":16013},[410],"height:1.1539em;vertical-align:-0.2997em;",[394,16015,16017,16020],{"className":16016},[4314],[394,16018,4361],{"className":16019,"style":4752},[4314,4359,4751],[394,16021,16023],{"className":16022},[423],[394,16024,16026,16046],{"className":16025},[427,913],[394,16027,16029,16043],{"className":16028},[431],[394,16030,16032],{"className":16031,"style":8807},[435],[394,16033,16034,16037],{"style":4768},[394,16035],{"className":16036,"style":443},[442],[394,16038,16040],{"className":16039},[447,448,449,450],[394,16041,5212],{"className":16042},[415,419,450],[394,16044,983],{"className":16045},[982],[394,16047,16049],{"className":16048},[431],[394,16050,16052],{"className":16051,"style":4787},[435],[394,16053],{},[394,16055],{"className":16056,"style":734},[465],[394,16058,16060,16063],{"className":16059},[415],[394,16061,384],{"className":16062},[415,419],[394,16064,16066],{"className":16065},[423],[394,16067,16069,16089],{"className":16068},[427,913],[394,16070,16072,16086],{"className":16071},[431],[394,16073,16075],{"className":16074,"style":5200},[435],[394,16076,16077,16080],{"style":5016},[394,16078],{"className":16079,"style":443},[442],[394,16081,16083],{"className":16082},[447,448,449,450],[394,16084,5212],{"className":16085},[415,419,450],[394,16087,983],{"className":16088},[982],[394,16090,16092],{"className":16091},[431],[394,16093,16095],{"className":16094,"style":5035},[435],[394,16096],{},[394,16098,16100,16103],{"className":16099},[415],[394,16101,12165],{"className":16102,"style":7567},[415,419],[394,16104,16106],{"className":16105},[423],[394,16107,16109,16129],{"className":16108},[427,913],[394,16110,16112,16126],{"className":16111},[431],[394,16113,16115],{"className":16114,"style":5200},[435],[394,16116,16117,16120],{"style":12305},[394,16118],{"className":16119,"style":443},[442],[394,16121,16123],{"className":16122},[447,448,449,450],[394,16124,5212],{"className":16125},[415,419,450],[394,16127,983],{"className":16128},[982],[394,16130,16132],{"className":16131},[431],[394,16133,16135],{"className":16134,"style":5035},[435],[394,16136],{},[394,16138,16140,16143],{"className":16139},[415],[394,16141,12165],{"className":16142,"style":7567},[415,419],[394,16144,16146],{"className":16145},[423],[394,16147,16149,16186],{"className":16148},[427,913],[394,16150,16152,16183],{"className":16151},[431],[394,16153,16155,16166],{"className":16154,"style":12826},[435],[394,16156,16157,16160],{"style":12829},[394,16158],{"className":16159,"style":443},[442],[394,16161,16163],{"className":16162},[447,448,449,450],[394,16164,5212],{"className":16165},[415,419,450],[394,16167,16168,16171],{"style":12841},[394,16169],{"className":16170,"style":443},[442],[394,16172,16174],{"className":16173},[447,448,449,450],[394,16175,16177,16180],{"className":16176},[415,450],[394,16178,457],{"className":16179},[415,450],[394,16181,461],{"className":16182},[415,450],[394,16184,983],{"className":16185},[982],[394,16187,16189],{"className":16188},[431],[394,16190,16192],{"className":16191,"style":12866},[435],[394,16193],{},[394,16195],{"className":16196,"style":7042},[465],[394,16198],{"className":16199,"style":626},[465],[394,16201,16203],{"className":16202},[516],[394,16204,16206],{"className":16205},[415],[394,16207,551],{"className":16208},[415,550],[394,16210],{"className":16211,"style":7042},[465],[394,16213],{"className":16214,"style":626},[465],[394,16216,16218,16221],{"className":16217},[406],[394,16219],{"className":16220,"style":871},[410],[394,16222,12165],{"className":16223,"style":7567},[415,419],", the inverse again coming\nfrom Fermat or the extended gcd.",[16226,16227,16230,16235],"section",{"className":16228,"dataFootnotes":376},[16229],"footnotes",[578,16231,16234],{"className":16232,"id":1615},[16233],"sr-only","Footnotes",[16236,16237,16238,16394,16571,16582],"ol",{},[14917,16239,16241,16244,16245,16386,16387],{"id":16240},"user-content-fn-clrs-count",[389,16242,16243],{},"CLRS",", Appendix C — Counting and Probability (§C.1): permutations, combinations, and the binomial coefficient ",[394,16246,16248],{"className":16247},[397],[394,16249,16251,16332,16374],{"className":16250,"ariaHidden":402},[401],[394,16252,16254,16257,16323,16326,16329],{"className":16253},[406],[394,16255],{"className":16256,"style":1497},[410],[394,16258,16260,16266,16317],{"className":16259},[415],[394,16261,16263],{"className":16262,"style":1226},[539,1225],[394,16264,540],{"className":16265},[1230,1507],[394,16267,16269],{"className":16268},[909],[394,16270,16272,16309],{"className":16271},[427,913],[394,16273,16275,16306],{"className":16274},[431],[394,16276,16278,16292],{"className":16277,"style":1520},[435],[394,16279,16280,16283],{"style":1523},[394,16281],{"className":16282,"style":443},[442],[394,16284,16286],{"className":16285},[447,448,449,450],[394,16287,16289],{"className":16288},[415,450],[394,16290,822],{"className":16291,"style":821},[415,419,450],[394,16293,16294,16297],{"style":1538},[394,16295],{"className":16296,"style":443},[442],[394,16298,16300],{"className":16299},[447,448,449,450],[394,16301,16303],{"className":16302},[415,450],[394,16304,605],{"className":16305},[415,419,450],[394,16307,983],{"className":16308},[982],[394,16310,16312],{"className":16311},[431],[394,16313,16315],{"className":16314,"style":1560},[435],[394,16316],{},[394,16318,16320],{"className":16319,"style":1226},[562,1225],[394,16321,563],{"className":16322},[1230,1507],[394,16324],{"className":16325,"style":466},[465],[394,16327,674],{"className":16328},[470],[394,16330],{"className":16331,"style":466},[465],[394,16333,16335,16338,16341,16344,16347,16350,16353,16356,16359,16362,16365,16368,16371],{"className":16334},[406],[394,16336],{"className":16337,"style":535},[410],[394,16339,605],{"className":16340},[415,419],[394,16342,667],{"className":16343},[562],[394,16345,11685],{"className":16346},[415],[394,16348,540],{"className":16349},[539],[394,16351,822],{"className":16352,"style":821},[415,419],[394,16354,667],{"className":16355},[562],[394,16357],{"className":16358,"style":734},[465],[394,16360,540],{"className":16361},[539],[394,16363,605],{"className":16364},[415,419],[394,16366],{"className":16367,"style":626},[465],[394,16369,457],{"className":16370},[516],[394,16372],{"className":16373,"style":626},[465],[394,16375,16377,16380,16383],{"className":16376},[406],[394,16378],{"className":16379,"style":535},[410],[394,16381,822],{"className":16382,"style":821},[415,419],[394,16384,15160],{"className":16385},[562],". ",[384,16388,16393],{"href":16389,"ariaLabel":16390,"className":16391,"dataFootnoteBackref":376},"#user-content-fnref-clrs-count","Back to reference 1",[16392],"data-footnote-backref","↩",[14917,16395,16397,16400,16401,16509,16510,16525,16526,16550,16551,16386,16566],{"id":16396},"user-content-fn-lucas",[389,16398,16399],{},"Skiena",", § — Combinatorics: Lucas' theorem reduces ",[394,16402,16404],{"className":16403},[397],[394,16405,16407,16500],{"className":16406,"ariaHidden":402},[401],[394,16408,16410,16413,16479,16482,16485,16494,16497],{"className":16409},[406],[394,16411],{"className":16412,"style":1497},[410],[394,16414,16416,16422,16473],{"className":16415},[415],[394,16417,16419],{"className":16418,"style":1226},[539,1225],[394,16420,540],{"className":16421},[1230,1507],[394,16423,16425],{"className":16424},[909],[394,16426,16428,16465],{"className":16427},[427,913],[394,16429,16431,16462],{"className":16430},[431],[394,16432,16434,16448],{"className":16433,"style":1520},[435],[394,16435,16436,16439],{"style":1523},[394,16437],{"className":16438,"style":443},[442],[394,16440,16442],{"className":16441},[447,448,449,450],[394,16443,16445],{"className":16444},[415,450],[394,16446,2068],{"className":16447,"style":2067},[415,419,450],[394,16449,16450,16453],{"style":1538},[394,16451],{"className":16452,"style":443},[442],[394,16454,16456],{"className":16455},[447,448,449,450],[394,16457,16459],{"className":16458},[415,450],[394,16460,605],{"className":16461},[415,419,450],[394,16463,983],{"className":16464},[982],[394,16466,16468],{"className":16467},[431],[394,16469,16471],{"className":16470,"style":1560},[435],[394,16472],{},[394,16474,16476],{"className":16475,"style":1226},[562,1225],[394,16477,563],{"className":16478},[1230,1507],[394,16480],{"className":16481,"style":7042},[465],[394,16483],{"className":16484,"style":626},[465],[394,16486,16488],{"className":16487},[516],[394,16489,16491],{"className":16490},[415],[394,16492,551],{"className":16493},[415,550],[394,16495],{"className":16496,"style":7042},[465],[394,16498],{"className":16499,"style":626},[465],[394,16501,16503,16506],{"className":16502},[406],[394,16504],{"className":16505,"style":4597},[410],[394,16507,381],{"className":16508},[415,419]," to a product of base-",[394,16511,16513],{"className":16512},[397],[394,16514,16516],{"className":16515,"ariaHidden":402},[401],[394,16517,16519,16522],{"className":16518},[406],[394,16520],{"className":16521,"style":4597},[410],[394,16523,381],{"className":16524},[415,419]," digit binomials when ",[394,16527,16529],{"className":16528},[397],[394,16530,16532],{"className":16531,"ariaHidden":402},[401],[394,16533,16535,16538,16541,16544,16547],{"className":16534},[406],[394,16536],{"className":16537,"style":761},[410],[394,16539,605],{"className":16540},[415,419],[394,16542,769],{"className":16543},[768],[394,16545],{"className":16546,"style":734},[465],[394,16548,2068],{"className":16549,"style":2067},[415,419]," exceed ",[394,16552,16554],{"className":16553},[397],[394,16555,16557],{"className":16556,"ariaHidden":402},[401],[394,16558,16560,16563],{"className":16559},[406],[394,16561],{"className":16562,"style":4597},[410],[394,16564,381],{"className":16565},[415,419],[384,16567,16393],{"href":16568,"ariaLabel":16569,"className":16570,"dataFootnoteBackref":376},"#user-content-fnref-lucas","Back to reference 2",[16392],[14917,16572,16574,16576,16577],{"id":16573},"user-content-fn-clrs-ie",[389,16575,16243],{},", Appendix C — Counting and Probability (§C.1): the inclusion–exclusion principle and the alternating-sign correction for unions. ",[384,16578,16393],{"href":16579,"ariaLabel":16580,"className":16581,"dataFootnoteBackref":376},"#user-content-fnref-clrs-ie","Back to reference 3",[16392],[14917,16583,16585,16587,16588,16743,16744],{"id":16584},"user-content-fn-clrs-crt",[389,16586,16243],{},", Ch. 31 — Number-Theoretic Algorithms (§31.5): the Chinese Remainder Theorem and the constructive ",[394,16589,16591],{"className":16590},[397],[394,16592,16594],{"className":16593,"ariaHidden":402},[401],[394,16595,16597,16600,16603,16606,16646,16686],{"className":16596},[406],[394,16598],{"className":16599,"style":12807},[410],[394,16601,4361],{"className":16602,"style":4752},[4314,4359,4751],[394,16604],{"className":16605,"style":734},[465],[394,16607,16609,16612],{"className":16608},[415],[394,16610,384],{"className":16611},[415,419],[394,16613,16615],{"className":16614},[423],[394,16616,16618,16638],{"className":16617},[427,913],[394,16619,16621,16635],{"className":16620},[431],[394,16622,16624],{"className":16623,"style":5200},[435],[394,16625,16626,16629],{"style":5016},[394,16627],{"className":16628,"style":443},[442],[394,16630,16632],{"className":16631},[447,448,449,450],[394,16633,5212],{"className":16634},[415,419,450],[394,16636,983],{"className":16637},[982],[394,16639,16641],{"className":16640},[431],[394,16642,16644],{"className":16643,"style":5035},[435],[394,16645],{},[394,16647,16649,16652],{"className":16648},[415],[394,16650,12165],{"className":16651,"style":7567},[415,419],[394,16653,16655],{"className":16654},[423],[394,16656,16658,16678],{"className":16657},[427,913],[394,16659,16661,16675],{"className":16660},[431],[394,16662,16664],{"className":16663,"style":5200},[435],[394,16665,16666,16669],{"style":12305},[394,16667],{"className":16668,"style":443},[442],[394,16670,16672],{"className":16671},[447,448,449,450],[394,16673,5212],{"className":16674},[415,419,450],[394,16676,983],{"className":16677},[982],[394,16679,16681],{"className":16680},[431],[394,16682,16684],{"className":16683,"style":5035},[435],[394,16685],{},[394,16687,16689,16692],{"className":16688},[415],[394,16690,12165],{"className":16691,"style":7567},[415,419],[394,16693,16695],{"className":16694},[423],[394,16696,16698,16735],{"className":16697},[427,913],[394,16699,16701,16732],{"className":16700},[431],[394,16702,16704,16715],{"className":16703,"style":12826},[435],[394,16705,16706,16709],{"style":12829},[394,16707],{"className":16708,"style":443},[442],[394,16710,16712],{"className":16711},[447,448,449,450],[394,16713,5212],{"className":16714},[415,419,450],[394,16716,16717,16720],{"style":12841},[394,16718],{"className":16719,"style":443},[442],[394,16721,16723],{"className":16722},[447,448,449,450],[394,16724,16726,16729],{"className":16725},[415,450],[394,16727,457],{"className":16728},[415,450],[394,16730,461],{"className":16731},[415,450],[394,16733,983],{"className":16734},[982],[394,16736,16738],{"className":16737},[431],[394,16739,16741],{"className":16740,"style":12866},[435],[394,16742],{}," formula. ",[384,16745,16393],{"href":16746,"ariaLabel":16747,"className":16748,"dataFootnoteBackref":376},"#user-content-fnref-clrs-crt","Back to reference 4",[16392],[16750,16751,16752],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}html.dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}",{"title":376,"searchDepth":18,"depth":18,"links":16754},[16755,16759,16760,16762,16763,16764,16765],{"id":580,"depth":18,"text":581,"children":16756},[16757,16758],{"id":2016,"depth":24,"text":2017},{"id":4217,"depth":24,"text":4218},{"id":4923,"depth":18,"text":4924},{"id":6959,"depth":18,"text":16761},"Computing (kn​)modp",{"id":9038,"depth":18,"text":9039},{"id":11694,"depth":18,"text":11695},{"id":14911,"depth":18,"text":14912},{"id":1615,"depth":18,"text":16234},"Modular exponentiation and, through Fermat's little\ntheorem, the modular inverse a−1≡ap−2(modp) are the hinge on\nwhich all of practical combinatorics swings: almost\nevery counting answer is a ratio of factorials, and a ratio modulo a prime is a\nproduct with an inverse. This lesson assembles the counting toolkit (permutations,\ncombinations, Pascal's rule, stars and bars) and then shows how to evaluate those\nquantities modulo a prime in constant time after a linear precompute. We finish\nwith two structural principles that recur everywhere: inclusion–exclusion for\ncounting unions, and the Chinese Remainder Theorem for stitching together\ncongruences.","md",{"moduleNumber":283,"lessonNumber":73,"order":16769},1004,true,[16772,16775,16778,16782],{"title":3662,"slug":16773,"difficulty":16774},"unique-paths","Medium",{"title":3658,"slug":16776,"difficulty":16777},"pascals-triangle-ii","Easy",{"title":16779,"slug":16780,"difficulty":16781},"Number of Music Playlists","number-of-music-playlists","Hard",{"title":8329,"slug":16783,"difficulty":16781},"count-anagrams","---\ntitle: Combinatorics & Counting\nmodule: Mathematical Algorithms\nmoduleNumber: 10\nlessonNumber: 4\norder: 1004\nsummary: >-\n  Counting is the arithmetic of finite sets. We build up from permutations\n  $n!$ and combinations $\\binom{n}{k}$, prove Pascal's rule by a bijection,\n  and count multisets with stars and bars. The practical core is computing\n  $\\binom{n}{k}\\bmod p$ in $O(1)$ from precomputed factorials and inverse\n  factorials. We close with inclusion–exclusion and the Chinese Remainder\n  Theorem, both of which lean on the modular inverse from the previous lesson.\ntopics: [Number Theory]\nsources:\n  - book: CLRS\n    ref: \"Appendix C — Counting and Probability\"\n  - book: Skiena\n    ref: \"§ — Combinatorics\"\n  - book: Erickson\n    ref: \"Ch. — (counting)\"\npractice:\n  - title: 'Unique Paths'\n    slug: unique-paths\n    difficulty: Medium\n  - title: \"Pascal's Triangle II\"\n    slug: pascals-triangle-ii\n    difficulty: Easy\n  - title: 'Number of Music Playlists'\n    slug: number-of-music-playlists\n    difficulty: Hard\n  - title: 'Count Anagrams'\n    slug: count-anagrams\n    difficulty: Hard\n---\n\n[Modular exponentiation](\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality) and, through Fermat's little\ntheorem, the **modular inverse** $a^{-1} \\equiv a^{p-2} \\pmod p$ are the hinge on\nwhich all of practical combinatorics swings: almost\nevery counting answer is a ratio of factorials, and a ratio modulo a prime is a\nproduct with an inverse. This lesson assembles the counting toolkit (permutations,\ncombinations, Pascal's rule, stars and bars) and then shows how to evaluate those\nquantities **modulo a prime** in constant time after a linear precompute. We finish\nwith two structural principles that recur everywhere: **inclusion–exclusion** for\ncounting unions, and the **Chinese Remainder Theorem** for stitching together\ncongruences.\n\n## Permutations and combinations\n\nA **permutation** is an ordering of distinct objects. There are $n$ choices for the\nfirst position, $n-1$ for the second, and so on, giving\n\n$$\nn! = n \\cdot (n-1) \\cdots 2 \\cdot 1, \\qquad 0! = 1.\n$$\n\nIf we order only $r$ of the $n$ objects, we stop the product after $r$ factors:\n\n$$\nnPr = \\frac{n!}{(n-r)!} = n(n-1)\\cdots(n-r+1).\n$$\n\nA **combination** counts _subsets_ of size $r$, where orderings no longer matter.\nEach $r$-subset can be ordered in $r!$ ways, so dividing $nPr$ by $r!$ removes the\novercount:\n\n$$\n\\binom{n}{r} = \\frac{nPr}{r!} = \\frac{n!}{r!\\,(n-r)!}.\n$$\n\nThe quantity $\\binom{n}{r}$, read \"$n$ choose $r$,\" is the **binomial coefficient**.[^clrs-count]\nIt is symmetric, $\\binom{n}{r} = \\binom{n}{n-r}$ (choosing which $r$ to include is the\nsame as choosing which $n-r$ to exclude), and the two boundary values are\n$\\binom{n}{0} = \\binom{n}{n} = 1$.\n\n### Pascal's rule\n\nBinomial coefficients satisfy a recurrence that lets us build them additively, with\nno division at all:\n\n> **Lemma (Pascal's rule).** For $0 \u003C k \u003C n$,\n> $$\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}.$$\n\n_Combinatorial proof._ Fix a distinguished element $n$. Every $k$-subset of\n$\\{1,\\dots,n\\}$ either **contains** $n$ or **does not**. Those that contain $n$ are\nformed by choosing the remaining $k-1$ elements from the other $n-1$, giving\n$\\binom{n-1}{k-1}$ of them. Those that omit $n$ choose all $k$ elements from the\nother $n-1$, giving $\\binom{n-1}{k}$. The two cases are disjoint and exhaustive,\nso their counts add. $\\qed$\n\nArranging these values in rows is **Pascal's triangle**: each interior entry is the\nsum of the two directly above it.\n\n$$\n% caption: Pascal's triangle — each cell $\\binom{n}{k}=\\binom{n-1}{k-1}+\\binom{n-1}{k}$\n\\begin{tikzpicture}[\n  every node\u002F.style={circle, draw, minimum size=8mm, inner sep=0, font=\\small},\n  x=11mm, y=12mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  % row 0\n  \\node (n00) at (0,0) {$1$};\n  % row 1\n  \\node (n10) at (-0.5,-1) {$1$};\n  \\node (n11) at (0.5,-1) {$1$};\n  % row 2\n  \\node (n20) at (-1,-2) {$1$};\n  \\node (n21) at (0,-2) {$2$};\n  \\node (n22) at (1,-2) {$1$};\n  % row 3\n  \\node (n30) at (-1.5,-3) {$1$};\n  \\node (n31) at (-0.5,-3) {$3$};\n  \\node (n32) at (0.5,-3) {$3$};\n  \\node (n33) at (1.5,-3) {$1$};\n  % row 4 - highlight the 6 = 3 + 3 cell\n  \\node (n40) at (-2,-4) {$1$};\n  \\node (n41) at (-1,-4) {$4$};\n  \\node[draw=acc, fill=acc!15, very thick] (n42) at (0,-4) {$6$};\n  \\node (n43) at (1,-4) {$4$};\n  \\node (n44) at (2,-4) {$1$};\n  % the two parents feeding the highlighted cell (derivation arrows in red)\n  \\draw[->, red!75!black, thick, shorten >=1pt] (n31) -- (n42);\n  \\draw[->, red!75!black, thick, shorten >=1pt] (n32) -- (n42);\n\\end{tikzpicture}\n$$\n\nThe highlighted cell is $\\binom{4}{2} = 6 = 3 + 3 = \\binom{3}{1} + \\binom{3}{2}$.\nBecause each entry needs only the row above, the whole triangle up to row $n$ is an\n$O(n^2)$ [dynamic program](\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples), the right approach when $n$ is small or when no modulus is\ninvolved (and the basis for _Pascal's Triangle II_ and grid-path problems like\n_Unique Paths_, whose answer is exactly $\\binom{m+n-2}{\\,m-1\\,}$).\n\nThat grid-path count is Pascal's rule in disguise: label each lattice node with the\nnumber of monotone (right\u002Fdown) paths reaching it, and each node is the sum of its\nleft and top neighbours, exactly the additive recurrence. On a $3\\times 3$ grid of\nnodes the corner reads $\\binom{4}{2}=6$.\n\n$$\n% caption: Lattice paths on a $3\\times 3$ grid: each node sums its left and top neighbour; corner is $\\binom{4}{2}=6$\n\\begin{tikzpicture}[every node\u002F.style={font=\\small}, x=14mm, y=14mm, >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  % grid edges as shortened unit segments so they sit in the gaps, clear of the digits\n  \\foreach \\i in {0,1,2}{\n    \\foreach \\j in {0,1}{\n      \\draw[black!30, shorten >=4mm, shorten \u003C=4mm] (\\j,-\\i) -- (\\j+1,-\\i);\n      \\draw[black!30, shorten >=4mm, shorten \u003C=4mm] (\\i,-\\j) -- (\\i,-\\j-1);\n    }\n  }\n  \\foreach \\c\u002F\\r\u002F\\v\u002F\\hl in {\n    0\u002F0\u002F1\u002F0, 1\u002F0\u002F1\u002F0, 2\u002F0\u002F1\u002F0,\n    0\u002F1\u002F1\u002F0, 1\u002F1\u002F2\u002F0, 2\u002F1\u002F3\u002F0,\n    0\u002F2\u002F1\u002F0, 1\u002F2\u002F3\u002F0, 2\u002F2\u002F6\u002F1}{\n    \\ifnum\\hl=1\n      \\node[draw=acc, circle, fill=acc!15, very thick, minimum size=8mm, inner sep=0] at (\\c,-\\r) {$\\v$};\n    \\else\n      \\node[draw, circle, fill=black!6, minimum size=7mm, inner sep=0] at (\\c,-\\r) {$\\v$};\n    \\fi\n  }\n  \\node[acc, font=\\footnotesize] at (1,-2.55) {start top-left, only right\u002Fdown moves};\n\\end{tikzpicture}\n$$\n\n### The binomial theorem\n\nThe coefficients earn their name from the expansion\n\n$$\n(x + y)^n = \\sum_{k=0}^{n} \\binom{n}{k}\\, x^{k}\\, y^{\\,n-k},\n$$\n\nsince each term of the product picks $x$ from $k$ of the $n$ factors and $y$ from the\nrest, and there are $\\binom{n}{k}$ ways to make that choice. Setting $x = y = 1$ gives\n$\\sum_k \\binom{n}{k} = 2^n$: the number of subsets of an $n$-set, counted by size.\n\n## Combinations with repetition: stars and bars\n\nHow many ways can we write a non-negative integer $n$ as an **ordered** sum of $k$\nnon-negative parts, $n = x_1 + x_2 + \\dots + x_k$ with each $x_i \\ge 0$? Equivalently,\nhow many multisets of size $n$ can we draw from $k$ distinct types?\n\n> **Lemma (Stars and bars).** The number of such solutions is\n> $$\\binom{n + k - 1}{\\,k - 1\\,}.$$\n\n_Bijection._ Lay out $n$ identical **stars** in a row and insert $k-1$ **bars** among\nthem; the bars cut the stars into $k$ ordered groups, the $i$-th group's size being\n$x_i$. For example with $n = 5$, $k = 3$:\n\n$$\n\\star\\,\\star \\mid\\; \\mid \\star\\,\\star\\,\\star \\quad\\longleftrightarrow\\quad\nx_1 = 2,\\; x_2 = 0,\\; x_3 = 3.\n$$\n\nEvery arrangement of $n$ stars and $k-1$ bars in a line of $n + k - 1$ symbols yields\nexactly one solution and vice versa, so the count is the number of ways to choose\nwhich $k-1$ of the $n + k - 1$ positions hold bars, namely $\\binom{n+k-1}{k-1}$.\n$\\qed$\n\nConcretely, with $n=5$ and $k=3$ there are $n + k - 1 = 7$ slots; choosing the $2$ of\nthem that hold bars fixes the three part sizes at once, so the count is\n$\\binom{7}{2}=21$.\n\n$$\n% caption: Stars and bars: $5$ stars and $2$ bars in $7$ slots encode $(x_1,x_2,x_3)=(2,0,3)$\n\\begin{tikzpicture}[every node\u002F.style={font=\\small}, x=9mm, y=9mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\foreach \\i in {1,...,7}{\n    \\draw (\\i-0.45,-0.45) rectangle (\\i+0.45,0.45);\n  }\n  \\foreach \\i in {1,2,5,6,7}{ \\node at (\\i,0) {$\\star$}; }\n  \\foreach \\i in {3,4}{ \\fill[acc!15] (\\i-0.45,-0.45) rectangle (\\i+0.45,0.45); \\draw[draw=acc, very thick] (\\i-0.45,-0.45) rectangle (\\i+0.45,0.45); \\node at (\\i,0) {$\\mid$}; }\n  \\draw[thick] (0.55,-0.65) -- (2.45,-0.65);\n  \\node[font=\\footnotesize] at (1.5,-1.1) {$x_1{=}2$};\n  \\node[font=\\footnotesize] at (3.5,-1.1) {$x_2{=}0$};\n  \\draw[thick] (4.55,-0.65) -- (7.45,-0.65);\n  \\node[font=\\footnotesize] at (6,-1.1) {$x_3{=}3$};\n  \\node[acc, font=\\footnotesize] at (4,1.25) {choose $2$ of $7$ slots for bars: $\\binom{7}{2}=21$};\n\\end{tikzpicture}\n$$\n\n## Computing $\\binom{n}{k} \\bmod p$\n\nCompetitive and large-scale problems ask for counts modulo a\nprime $p$ (typically $10^9 + 7$) because the true values are astronomically large.\nThe factorial formula has a division by $r!\\,(n-r)!$, and **division is not defined\nmodulo $p$**; what stands in for it is multiplication by a [modular inverse](\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics). Since $p$ is prime,\nFermat gives $a^{-1} \\equiv a^{p-2} \\pmod p$ for any $a \\not\\equiv 0$, computed by the\nmodular exponentiation routine.\n\nThe plan: precompute the factorials $\\text{fact}[i] = i! \\bmod p$ for all $i \\le N$,\nand the **inverse factorials** $\\text{invfact}[i] = (i!)^{-1} \\bmod p$. With both\ntables in hand, every binomial coefficient is a single product.\n\n```algorithm\ncaption: $\\textsc{Precompute-Factorials}(N, p)$ — $O(N)$ tables for $O(1)$ queries\n$\\text{fact}[0] \\gets 1$\nfor $i \\gets 1$ to $N$ do\n  $\\text{fact}[i] \\gets \\text{fact}[i-1] \\cdot i \\bmod p$\n$\\text{invfact}[N] \\gets \\textsc{Mod-Pow}(\\text{fact}[N],\\, p-2,\\, p)$ \u002F\u002F one Fermat inverse\nfor $i \\gets N$ downto $1$ do\n  $\\text{invfact}[i-1] \\gets \\text{invfact}[i] \\cdot i \\bmod p$ \u002F\u002F peel a factor\n```\n\nThe downward loop is the trick that keeps the precompute at $O(N)$ rather than\n$O(N\\log p)$: only **one** modular exponentiation is needed, for $\\text{invfact}[N]$;\neach smaller inverse factorial follows from $(i-1)!^{-1} = i!^{-1}\\cdot i$. Then each\nquery is constant time:\n\n$$\n\\binom{n}{k} \\equiv \\text{fact}[n]\\cdot\\text{invfact}[k]\\cdot\\text{invfact}[n-k] \\pmod p,\n\\qquad 0 \\le k \\le n \\le N.\n$$\n\nThis $O(N)$-precompute, $O(1)$-query scheme is exactly what _Number of Music\nPlaylists_ and _Count Anagrams_ need, since both reduce to products and ratios of\nfactorials modulo $10^9 + 7$.\n\n> **Theorem (Lucas').** When $n$ and $k$ can exceed $p$ itself, the factorial tables\n> break (they contain factors of $p$, hence zeros). Lucas reduces the problem to\n> digits in base $p$: writing $n = \\sum n_i p^i$ and $k = \\sum k_i p^i$,\n> $\\binom{n}{k} \\equiv \\prod_i \\binom{n_i}{k_i} \\pmod p$, a product of small binomial\n> coefficients each computable from a table of size $p$.[^lucas]\n\n## Inclusion–exclusion\n\nTo count a **union** of overlapping sets we cannot simply add their sizes, since elements\nin several sets get counted several times. Inclusion–exclusion corrects the overcount\nwith alternating signs:\n\n$$\n\\Bigl|\\,\\bigcup_{i=1}^{n} A_i\\,\\Bigr| =\n\\sum_i |A_i| \\;-\\; \\sum_{i\u003Cj} |A_i \\cap A_j| \\;+\\; \\sum_{i\u003Cj\u003Ck} |A_i \\cap A_j \\cap A_k|\n\\;-\\; \\cdots \\;+\\; (-1)^{n+1}\\Bigl|\\bigcap_i A_i\\Bigr|.\n$$\n\n> **Lemma (Alternating-sign principle).** Add the sizes of all single sets, subtract all\n> pairwise intersections, add all triple intersections, and so on. An element lying\n> in exactly $m$ of the sets is counted $\\sum_{j=1}^{m}(-1)^{j+1}\\binom{m}{j} = 1$\n> time in total, exactly once, as it should be.[^clrs-ie]\n\n$$\n% caption: Inclusion–exclusion — add singles, subtract pairs, add the triple (in acc)\n\\begin{tikzpicture}[\n  every node\u002F.style={font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % three overlapping circles\n  \\draw[thick] (0,0) circle (1.5cm);\n  \\draw[thick] (2,0) circle (1.5cm);\n  \\draw[thick] (1,1.7) circle (1.5cm);\n  % triple overlap marked with a small dot, label pulled below-right with a leader\n  \\fill[acc] (1,0.57) circle (1.3pt);\n  \\draw[acc, thin] (1,0.57) .. controls (2.7,-0.6) .. (3.2,-2.0);\n  \\node[acc, right] at (3.2,-2.05) {$+\\,|A\\cap B\\cap C|$};\n  % set labels\n  \\node at (-1.05,-1.15) {$A$};\n  \\node at (3.05,-1.15) {$B$};\n  \\node at (1,3.15) {$C$};\n  % singles\n  \\node at (-0.65,-0.15) {$+A$};\n  \\node at (2.65,-0.15) {$+B$};\n  \\node at (1,2.45) {$+C$};\n  % pairwise overlaps\n  \\node at (1,-0.15) {$-$};\n  \\node at (-0.05,1.0) {$-$};\n  \\node at (2.05,1.0) {$-$};\n\\end{tikzpicture}\n$$\n\n**Worked example (counting coprime-to-a-set integers).** How many integers in\n$[1, 30]$ are divisible by none of $\\{2, 3, 5\\}$? Let $A, B, C$ be the multiples of\n$2, 3, 5$ respectively. Then $|A| = 15,\\ |B| = 10,\\ |C| = 6$;\n$|A\\cap B| = \\lfloor 30\u002F6\\rfloor = 5,\\ |A\\cap C| = 3,\\ |B\\cap C| = 2$; and\n$|A\\cap B\\cap C| = \\lfloor 30\u002F30\\rfloor = 1$. So\n\n$$\n|A\\cup B\\cup C| = (15+10+6) - (5+3+2) + 1 = 22,\n$$\n\nleaving $30 - 22 = 8$ integers divisible by none of $2,3,5$, which are exactly\n$1, 7, 11, 13, 17, 19, 23, 29$. The same alternating sum, applied with $A_i$ = \"maps\nposition $i$ to itself,\" counts **derangements** $D_n = n!\\sum_{j=0}^{n}(-1)^j\u002Fj!$.\n\n## The Chinese Remainder Theorem\n\nInclusion–exclusion combines counts; the **Chinese Remainder Theorem** (CRT)\ncombines _congruences_. Given a system\n\n$$\nx \\equiv a_1 \\pmod{m_1},\\quad x \\equiv a_2 \\pmod{m_2},\\quad\\dots,\\quad x \\equiv a_n \\pmod{m_n}\n$$\n\nwith the moduli **pairwise coprime**, CRT guarantees a **unique** solution modulo\n$M = \\prod_i m_i$.[^clrs-crt] The construction is explicit and again uses the modular\ninverse. Let $M_i = M \u002F m_i = \\prod_{j\\ne i} m_j$. Because the $m_j$ are coprime to\n$m_i$, so is $M_i$, hence $M_i$ has an inverse modulo $m_i$; call it\n$M_i^{-1} \\equiv M_i^{\\,\\varphi(m_i)-1}$ (or $M_i^{m_i-2}$ when $m_i$ is prime). Then\n\n$$\nx \\equiv \\sum_{i=1}^{n} a_i\\, M_i\\, M_i^{-1} \\pmod{M}.\n$$\n\nEach term $a_i M_i M_i^{-1}$ is $\\equiv a_i \\pmod{m_i}$ (since $M_i M_i^{-1}\\equiv 1$\nthere) and $\\equiv 0$ modulo every other $m_j$ (since $m_j \\mid M_i$), so the sum\nsatisfies all $n$ congruences simultaneously. Concretely, to solve\n$x\\equiv 2\\,(3)$ and $x\\equiv 3\\,(5)$ each term acts as a **selector**: one lands\non its own residue and vanishes modulo the other, so adding them assembles the\nanswer $x\\equiv 8\\pmod{15}$ one congruence at a time.\n\n$$\n% caption: CRT as a sum of selectors: each $a_iM_iM_i^{-1}$ hits its own residue and is $0$ modulo the other, so the terms add to $x\\equiv 8\\pmod{15}$.\n\\begin{tikzpicture}[every node\u002F.style={font=\\small}, x=1cm, y=1cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-3.6,1.2) rectangle (4.0,-3.5);\n  % column headers\n  \\node[font=\\footnotesize] at (1.4,0.7) {$\\bmod 3$};\n  \\node[font=\\footnotesize] at (3.0,0.7) {$\\bmod 5$};\n  \\draw[thick] (-3.4,0.35) -- (3.7,0.35);\n  % row 1: term for m=3\n  \\node[anchor=west] at (-3.4,0) {$a_1M_1M_1^{-1}=20$};\n  \\node[acc] at (1.4,0) {$2$};\n  \\node[black!55] at (3.0,0) {$0$};\n  % row 2: term for m=5\n  \\node[anchor=west] at (-3.4,-0.9) {$a_2M_2M_2^{-1}=18$};\n  \\node[black!55] at (1.4,-0.9) {$0$};\n  \\node[acc] at (3.0,-0.9) {$3$};\n  \\draw (-3.4,-1.35) -- (3.7,-1.35);\n  % sum row\n  \\node[anchor=west] at (-3.4,-1.8) {sum $=38\\equiv 8\\ (15)$};\n  \\node[draw=acc, fill=acc!15, very thick, minimum size=7mm, inner sep=1pt] at (1.4,-1.8) {$2$};\n  \\node[draw=acc, fill=acc!15, very thick, minimum size=7mm, inner sep=1pt] at (3.0,-1.8) {$3$};\n  \\node[acc, font=\\footnotesize, align=center] at (0.1,-2.85)\n    {each term is a selector: $1$ on its own modulus, $0$ on the other};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{CRT}(a[\\,], m[\\,])$ — combine congruences with pairwise-coprime moduli\n$M \\gets \\prod_i m_i$\n$x \\gets 0$\nfor $i \\gets 1$ to $n$ do\n  $M_i \\gets M \u002F m_i$\n  $y_i \\gets \\textsc{Mod-Inverse}(M_i \\bmod m_i,\\; m_i)$\n  $x \\gets (x + a_i \\cdot M_i \\cdot y_i) \\bmod M$\nreturn $x$\n```\n\nCRT is what lets us compute modulo a large composite $M$ by working independently in\neach prime-power factor and reassembling, the structural twin of how stars-and-bars\nor inclusion–exclusion decompose a hard count into manageable pieces.\n\n## Takeaways\n\n- **Permutations** count orderings ($n!$, or $nPr = n!\u002F(n-r)!$); **combinations**\n  count subsets, $\\binom{n}{r} = n!\u002F(r!\\,(n-r)!)$, dividing out the $r!$ orderings.\n- **Pascal's rule** $\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}$ (element $n$ in\n  or out) builds the triangle additively in $O(n^2)$ with no division.\n- **Stars and bars**: the ordered non-negative solutions of $x_1+\\dots+x_k = n$ number\n  $\\binom{n+k-1}{k-1}$, via the bars-between-stars bijection.\n- To compute $\\binom{n}{k}\\bmod p$, precompute **factorials** and **inverse\n  factorials** (one Fermat inverse, then peel factors) in $O(n)$, giving $O(1)$ per\n  query; use **Lucas' theorem** when $n, k > p$.\n- **Inclusion–exclusion** counts unions by alternating add\u002Fsubtract over all\n  intersections; the alternating signs make each element net-counted exactly once.\n- The **Chinese Remainder Theorem** uniquely solves a system of congruences with\n  coprime moduli via $x = \\sum_i a_i M_i M_i^{-1} \\bmod M$, the inverse again coming\n  from Fermat or the extended gcd.\n\n[^clrs-count]: **CLRS**, Appendix C — Counting and Probability (§C.1): permutations, combinations, and the binomial coefficient $\\binom{n}{r} = n!\u002F(r!\\,(n-r)!)$.\n[^clrs-ie]: **CLRS**, Appendix C — Counting and Probability (§C.1): the inclusion–exclusion principle and the alternating-sign correction for unions.\n[^clrs-crt]: **CLRS**, Ch. 31 — Number-Theoretic Algorithms (§31.5): the Chinese Remainder Theorem and the constructive $\\sum a_i M_i M_i^{-1}$ formula.\n[^lucas]: **Skiena**, § — Combinatorics: Lucas' theorem reduces $\\binom{n}{k}\\bmod p$ to a product of base-$p$ digit binomials when $n, k$ exceed $p$.\n",{"text":16786,"minutes":16787,"time":16788,"words":16789},"7 min read",6.31,378600,1262,{"title":327,"description":16766},[16792,16794,16796],{"book":16243,"ref":16793},"Appendix C — Counting and Probability",{"book":16399,"ref":16795},"§ — Combinatorics",{"book":16797,"ref":16798},"Erickson","Ch. — (counting)","available","01.algorithms\u002F10.mathematical-algorithms\u002F04.combinatorics",[314],"PILcqqF3oncF1pV0_hTmyD7SRLn12JGZhGrl74FwM7Q",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":16804,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":16805,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":16806,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":16807,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":16808,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":16809,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":16810,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":16811,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":16812,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":16813,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":16814,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":16815,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":16816,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":16817,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":16818,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":16819,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":16820,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":16821,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":16822,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":16823,"\u002Falgorithms\u002Fsequences\u002Ftries":16824,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":16825,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":16826,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":16827,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":16828,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":16829,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":16830,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":16831,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":16832,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":16833,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":16834,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":16835,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":16836,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":16837,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":16838,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":16839,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":16840,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":16841,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":16842,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":16843,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":16844,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":16845,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":16846,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":16847,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":16848,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":16849,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":16850,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":16820,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":16851,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":16852,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":16789,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":16853,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":16836,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":16854,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":16855,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":16816,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":16856,"\u002Falgorithms":16857,"\u002Ftheory-of-computation":16858,"\u002Fcomputer-architecture":16858,"\u002Fphysical-computing":16858,"\u002Fdatabases":16858,"\u002Fdeep-learning":16858},1763,2107,1738,2628,1723,2048,1697,1044,1542,1565,1679,1586,1388,1465,1971,1455,1533,1483,1578,1791,1481,2704,1658,2070,1978,2080,1568,1451,1291,1543,1883,1443,1599,2038,2241,1744,1678,2288,1929,1657,1412,1554,1418,1713,1798,1694,1762,1534,1595,1495,1630,2306,2142,107,0,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":16860,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":16861,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":16862,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":16863,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":16864,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":16865,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":16866,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":16867,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":16868,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":16869,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":16870,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":16871,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":16872,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":16873,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":16874,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":16875,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":16876,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":16877,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":16878,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":16879,"\u002Falgorithms\u002Fsequences\u002Ftries":16880,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":16881,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":16882,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":16883,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":16884,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":16885,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":16886,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":16887,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":16888,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":16889,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":16890,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":16891,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":16892,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":16893,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":16894,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":16895,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":16896,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":16897,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":16898,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":16899,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":16900,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":16901,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":16902,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":16903,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":16904,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":16905,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":16906,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":16907,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":16908,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":16909,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":16910,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":16911,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":16912,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":16913,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":16914,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":16915,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":16916,"\u002Falgorithms":16917,"\u002Ftheory-of-computation":16920,"\u002Fcomputer-architecture":16923,"\u002Fphysical-computing":16926,"\u002Fdatabases":16929,"\u002Fdeep-learning":16932},{"path":11,"title":10,"module":5,"summary":14},{"path":17,"title":16,"module":5,"summary":20},{"path":23,"title":22,"module":5,"summary":27},{"path":34,"title":33,"module":29,"summary":37},{"path":40,"title":39,"module":29,"summary":43},{"path":46,"title":45,"module":29,"summary":49},{"path":56,"title":55,"module":51,"summary":59},{"path":62,"title":61,"module":51,"summary":64},{"path":67,"title":66,"module":51,"summary":70},{"path":78,"title":77,"module":72,"summary":81},{"path":84,"title":83,"module":72,"summary":87},{"path":90,"title":89,"module":72,"summary":92},{"path":95,"title":94,"module":72,"summary":98},{"path":101,"title":100,"module":72,"summary":104},{"path":107,"title":106,"module":72,"summary":112},{"path":115,"title":114,"module":72,"summary":119},{"path":126,"title":125,"module":121,"summary":129},{"path":132,"title":131,"module":121,"summary":134},{"path":137,"title":136,"module":121,"summary":140},{"path":143,"title":142,"module":121,"summary":146},{"path":149,"title":148,"module":121,"summary":151},{"path":158,"title":157,"module":153,"summary":162},{"path":165,"title":164,"module":153,"summary":167},{"path":170,"title":169,"module":153,"summary":172},{"path":175,"title":174,"module":153,"summary":177},{"path":180,"title":179,"module":153,"summary":182},{"path":185,"title":184,"module":153,"summary":187},{"path":190,"title":189,"module":153,"summary":192},{"path":195,"title":194,"module":153,"summary":198},{"path":201,"title":200,"module":153,"summary":204},{"path":211,"title":210,"module":206,"summary":213},{"path":216,"title":215,"module":206,"summary":219},{"path":222,"title":221,"module":206,"summary":224},{"path":227,"title":226,"module":206,"summary":229},{"path":236,"title":235,"module":231,"summary":238},{"path":241,"title":240,"module":231,"summary":244},{"path":247,"title":246,"module":231,"summary":249},{"path":252,"title":251,"module":231,"summary":254},{"path":257,"title":256,"module":231,"summary":259},{"path":262,"title":261,"module":231,"summary":264},{"path":267,"title":266,"module":231,"summary":269},{"path":272,"title":271,"module":231,"summary":274},{"path":277,"title":276,"module":231,"summary":279},{"path":282,"title":281,"module":231,"summary":285},{"path":292,"title":291,"module":287,"summary":295},{"path":298,"title":297,"module":287,"summary":300},{"path":303,"title":302,"module":287,"summary":305},{"path":312,"title":311,"module":307,"summary":315},{"path":318,"title":317,"module":307,"summary":320},{"path":323,"title":322,"module":307,"summary":325},{"path":328,"title":327,"module":307,"summary":330},{"path":338,"title":337,"module":332,"summary":341},{"path":344,"title":343,"module":332,"summary":346},{"path":349,"title":348,"module":332,"summary":351},{"path":359,"title":358,"module":353,"summary":362},{"path":364,"title":361,"module":353,"summary":366},{"path":369,"title":368,"module":353,"summary":373},{"path":16918,"title":16919,"module":376,"summary":376},"\u002Falgorithms","Algorithms",{"path":16921,"title":16922,"module":376,"summary":376},"\u002Ftheory-of-computation","Theory of Computation",{"path":16924,"title":16925,"module":376,"summary":376},"\u002Fcomputer-architecture","Computer Architecture",{"path":16927,"title":16928,"module":376,"summary":376},"\u002Fphysical-computing","Physical Computing",{"path":16930,"title":16931,"module":376,"summary":376},"\u002Fdatabases","Databases",{"path":16933,"title":16934,"module":376,"summary":376},"\u002Fdeep-learning","Deep Learning",1781526660294]