{"history":[],"article":{"id":"d74c0244-d75d-43dc-8725-d59b3f0cd85a","title":"Binomial coefficient","slug":"binomial-coefficient","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **binomial [coefficients](https://bliptext.com/articles/coefficient)** are the positive [integers](https://bliptext.com/articles/integer) that occur as [coefficients](https://bliptext.com/articles/coefficient) in the [binomial theorem](https://bliptext.com/articles/binomial-theorem).\nCommonly, a binomial coefficient is indexed by a pair of integers\nn ≥ k ≥ 0\nand is written It is the coefficient of the\nxk\nterm in the [polynomial expansion](https://bliptext.com/articles/polynomial-expansion) of the [binomial](https://bliptext.com/articles/binomial-polynomial) [power](https://bliptext.com/articles/exponentiation)\n(1 + x)n\nwhich using [factorial](https://bliptext.com/articles/factorial) notation can be compactly expressed as\nFor example, the fourth power of\n1 + x\nis and the binomial coefficient is the coefficient of the\nx2\nterm.\nArranging the numbers in successive rows for\nn = 0, 1, 2, ...\ngives a triangular array called [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), satisfying the [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nThe binomial coefficients occur in many areas of mathematics, and especially in [combinatorics](https://bliptext.com/articles/combinatorics).\nIn combinatorics the symbol is usually read as \"n choose k\" because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements.\nFor example, there are ways to choose\n2\nelements from\n, namely\n,\n,\n,\n,\nand\n.\nThe first form of the binomial coefficients can be generalized to for any [complex number](https://bliptext.com/articles/complex-number) z and integer\nk ≥ 0\n, and many of their properties continue to hold in this more general form.\n# [The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. | upload.wikimedia.org/wikipedia/commons/f/f6/Pascal%27s///triangle///5.svg]\n# [Visualisation of binomial expansion up to the 4th power | upload.wikimedia.org/wikipedia/commons/7/7c/binomial///theorem///visualisation.svg]\n\n# History and notation\n[Andreas von Ettingshausen](https://bliptext.com/articles/andreas-von-ettingshausen) introduced the notation \\tbinom nk in 1826, although the numbers were known centuries earlier (see [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle)).\nIn about 1150, the Indian mathematician [Bhaskaracharya](https://bliptext.com/articles/bhaskaracharya) gave an exposition of binomial coefficients in his book [Līlāvatī](https://bliptext.com/articles/l-l-vat).\nAlternative notations include\nC(n, k)\n,\nnCk\n,\nnCk\n,\nCk n\n,\nCn k\n, and\nCn,k\n, in all of which the C stands for [combinations](https://bliptext.com/articles/combination) or choices; the C notation means the number of ways to choose k out of n objects.\nMany calculators use variants of the C notation because they can represent it on a single-line display.\nIn this form the binomial coefficients are easily compared to the numbers of [k-permutations of n](https://bliptext.com/articles/permutation), written as\nP(n, k)\n, etc.\n\n# Definition and interpretations\nFor [natural numbers](https://bliptext.com/articles/natural-number) (taken to include 0) n and k, the binomial [coefficient](https://bliptext.com/articles/coefficient) \\tbinom nk can be defined as the [coefficient](https://bliptext.com/articles/coefficient) of the [monomial](https://bliptext.com/articles/monomial)\nXk\nin the expansion of\n(1 + X)n\n. The same coefficient also occurs (if\nk ≤ n\n) in the [binomial formula](https://bliptext.com/articles/binomial-formula)\n(valid for any elements x, y of a [commutative ring](https://bliptext.com/articles/commutative-ring)),\nwhich explains the name \"binomial coefficient\".\nAnother occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-[combinations](https://bliptext.com/articles/combination)) of an n-element set.\nThis number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power\n(1 + X)n\none temporarily labels the term X with an index i (running from\n1\nto n), then each subset of k indices gives after expansion a contribution\nXk\n, and the coefficient of that monomial in the result will be the number of such subsets.\nThis shows in particular that \\tbinom nk is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n [bits](https://bliptext.com/articles/bit) (digits 0 or 1) whose sum is k is given by \\tbinom nk, while the number of ways to write where every\nai\nis a nonnegative integer is given by \\tbinom{n+k-1}{n-1}.\nMost of these interpretations can be shown to be equivalent to counting k-combinations.\n\n# Computing the value of binomial coefficients\nSeveral methods exist to compute the value of without actually expanding a binomial power or counting k-combinations.\n\n## Recursive formula\nOne method uses the [recursive](https://bliptext.com/articles/recursion), purely additive formula\nfor all integers n,k such that\nwith boundary values\nfor all integers\nn ≥ 0\n.\nThe formula follows from considering the set\nand counting separately (a) the k-element groupings that include a particular set element, say \"i\", in every group (since \"i\" is already chosen to fill one spot in every group, we need only choose\nk − 1\nfrom the remaining\nn − 1\n) and (b) all the k-groupings that don't include \"i\"; this enumerates all the possible k-combinations of n elements. It also follows from tracing the contributions to Xk in\n(1 + X)n−1(1 + X)\n. As there is zero\nXn+1\nor\nX−1\nin\n(1 + X)n\n, one might extend the definition beyond the above boundaries to include when either\nk > n\nor\nk < 0\n. This recursive formula then allows the construction of [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), surrounded by white spaces where the zeros, or the trivial coefficients, would be.\n\n## Multiplicative formula\nA more efficient method to compute individual binomial coefficients is given by the formula\nwhere the numerator of the first fraction,, is a [falling factorial](https://bliptext.com/articles/falling-factorial).\nThis formula is easiest to understand for the combinatorial interpretation of binomial coefficients.\nThe numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects.\nThe denominator counts the number of distinct sequences that define the same k-combination when order is disregarded.\nThis formula can also be stated in a recursive form.\nUsing the \"C\" notation from above,, where C_{n,0} = 1.\nIt is readily derived by evaluating and can intuitively be understood as starting at the leftmost coefficient of the n-th row of [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), whose value is always 1, and recursively computing the next coefficient to its right until the k-th one is reached.\nDue to the symmetry of the [binomial coefficients](https://bliptext.com/articles/pascal-s-triangle) with regard to k and\nn − k\n, calculation of the above product, as well as the recursive relation, may be optimised by setting its upper limit to the smaller of k and\nn − k\n.\n\n## Factorial formula\nFinally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar [factorial](https://bliptext.com/articles/factorial) function:\nwhere\nn!\ndenotes the factorial of n. This formula follows from the multiplicative formula above by multiplying numerator and denominator by\n(n − k)!\nwhich leads to a more efficient multiplicative computational routine.\nUsing the [falling factorial notation](https://bliptext.com/articles/pochhammer-symbol),\n\n## Generalization and connection to the binomial series\nThe multiplicative formula allows the definition of binomial coefficients to be extended by replacing n by an arbitrary number α (negative, real, complex) or even an element of any [commutative ring](https://bliptext.com/articles/commutative-ring) in which all positive integers are invertible:\nWith this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the binomial coefficients:\nThis formula is valid for all complex numbers α and X with |X| < 1.\nIt can also be interpreted as an identity of [formal power series](https://bliptext.com/articles/formal-power-series) in X, where it actually can serve as definition of arbitrary powers of [power series](https://bliptext.com/articles/power-series) with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for [exponentiation](https://bliptext.com/articles/exponentiation), notably\nIf α is a nonnegative integer n, then all terms with\nk > n\nare zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula.\nHowever, for other values of α, including negative integers and rational numbers, the series is really infinite.\n\n# Pascal's triangle\n[Pascal's rule](https://bliptext.com/articles/pascal-s-rule) is the important [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nwhich can be used to prove by [mathematical induction](https://bliptext.com/articles/mathematical-induction) that \\tbinom n k is a natural number for all integer n ≥ 0 and all integer k, a fact that is not immediately obvious from [formula (1)](https://bliptext.com/articles/).\nTo the left and right of Pascal's triangle, the entries (shown as blanks) are all zero.\nPascal's rule also gives rise to [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle):\nRow number n contains the numbers for\nk = 0, …, n\n. It is constructed by first placing 1s in the outermost positions, and then filling each inner position with the sum of the two numbers directly above.\nThis method allows the quick calculation of binomial coefficients without the need for fractions or multiplications.\nFor instance, by looking at row number 5 of the triangle, one can quickly read off that\n# [1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence. | upload.wikimedia.org/wikipedia/commons/c/c1/Pascal%27s///triangle///-///1000th///row.png]\n\n# Combinatorics and statistics\nBinomial coefficients are of importance in [combinatorics](https://bliptext.com/articles/combinatorics) because they provide ready formulas for certain frequent counting problems:\n\n# Binomial coefficients as polynomials\nFor any nonnegative integer k, the expression can be written as a polynomial with denominator\nk!\nthis presents a [polynomial](https://bliptext.com/articles/polynomial) in t with [rational](https://bliptext.com/articles/rational-number) coefficients.\nAs such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments.\nThese \"generalized binomial coefficients\" appear in [Newton's generalized binomial theorem](https://bliptext.com/articles/binomial-theorem).\nFor each k, the polynomial can be characterized as the unique degree k polynomial\np(t)\nsatisfying\np(0) = p(1) = ⋯ = p(k − 1) = 0\nand\np(k) = 1\n.\nIts coefficients are expressible in terms of [Stirling numbers of the first kind](https://bliptext.com/articles/stirling-numbers-of-the-first-kind):\nThe [derivative](https://bliptext.com/articles/derivative) of can be calculated by [logarithmic differentiation](https://bliptext.com/articles/logarithmic-differentiation):\nThis can cause a problem when evaluated at integers from 0 to t-1, but using identities below we can compute the derivative as:\n\n## Binomial coefficients as a basis for the space of polynomials\nOver any [field](https://bliptext.com/articles/field-mathematics) of [characteristic 0](https://bliptext.com/articles/characteristic-algebra) (that is, any [field](https://bliptext.com/articles/field-mathematics) that contains the [rational numbers](https://bliptext.com/articles/rational-number)), each polynomial p(t) of degree at most d is uniquely expressible as a linear combination of binomial coefficients, because the binomial coefficients consist of one polynomial of each degree.\nThe coefficient ak is the kth difference of the sequence p(0), p(1), ..., p(k).\nExplicitly,\n\n## Integer-valued polynomials\nEach polynomial is [integer-valued](https://bliptext.com/articles/integer-valued-polynomial): it has an integer value at all integer inputs t. (One way to prove this is by induction on k using [Pascal's identity](https://bliptext.com/articles/pascal-s-identity).) Therefore, any integer linear combination of binomial coefficient polynomials is [integer-valued](https://bliptext.com/articles/integer-valued-polynomial) too.\nConversely, shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials.\nMore generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials.\n\n## Example\nThe integer-valued polynomial\n3t(3t + 1) / 2\ncan be rewritten as\n\n# Identities involving binomial coefficients\nThe factorial formula facilitates relating nearby binomial coefficients.\nFor instance, if k is a positive integer and n is arbitrary, then\nand, with a little more work,\nWe can also get\nMoreover, the following may be useful:\nFor constant n, we have the following recurrence:\nTo sum up, we have\n\n## Sums of the binomial coefficients\nThe formula\nsays that the elements in the nth row of Pascal's triangle always add up to 2 raised to the nth power.\nThis is obtained from the binomial theorem by setting\nx = 1\nand\ny = 1\n. The formula also has a natural combinatorial interpretation: the left side sums the number of subsets of {1, ..., n} of sizes k = 0, 1, ..., n, giving the total number of subsets.\n(That is, the left side counts the [power set](https://bliptext.com/articles/power-set) of {1, ..., n}.) However, these subsets can also be generated by successively choosing or excluding each element 1, ..., n; the n independent binary choices (bit-strings) allow a total of 2^n choices.\nThe left and right sides are two ways to count the same collection of subsets, so they are equal.\nThe formulas\nand\nfollow from the binomial theorem after [differentiating](https://bliptext.com/articles/derivative) with respect to x (twice for the latter) and then substituting\nx = y = 1\n.\nThe [Chu–Vandermonde identity](https://bliptext.com/articles/chu-vandermonde-identity), which holds for any complex values m and n and any non-negative integer k, is\nand can be found by examination of the coefficient of x^k in the expansion of\n(1 + x)m(1 + x)n−m = (1 + x)n\nusing equation.\nWhen\nm = 1\n, equation reduces to equation.\nIn the special case\nn = 2m, k = m\n, using, the expansion becomes (as seen in Pascal's triangle at right)\nwhere the term on the right side is a [central binomial coefficient](https://bliptext.com/articles/central-binomial-coefficient).\nAnother form of the Chu–Vandermonde identity, which applies for any integers j, k, and n satisfying\n0 ≤ j ≤ k ≤ n\n, is\nThe proof is similar, but uses the binomial series expansion with negative integer exponents.\nWhen\nj = k\n, equation gives the [hockey-stick identity](https://bliptext.com/articles/hockey-stick-identity)\nand its relative\nLet F(n) denote the n-th [Fibonacci number](https://bliptext.com/articles/fibonacci-number).\nThen\nThis can be proved by [induction](https://bliptext.com/articles/mathematical-induction) using or by [Zeckendorf's representation](https://bliptext.com/articles/zeckendorf-s-theorem).\nA combinatorial proof is given below.\n\n### Multisections of sums\nFor integers s and t such that [series multisection](https://bliptext.com/articles/series-multisection) gives the following identity for the sum of binomial coefficients:\nFor small s, these series have particularly nice forms; for example,\n\n### Partial sums\nAlthough there is no [closed formula](https://bliptext.com/articles/closed-formula) for [partial sums](https://bliptext.com/articles/partial-sum)\nof binomial coefficients, one can again use and induction to show that for\nk = 0, …, n − 1\n, with special case\nfor\nn > 0\n. This latter result is also a special case of the result from the theory of [finite differences](https://bliptext.com/articles/finite-differences) that for any polynomial P(x) of degree less than n,\nDifferentiating k times and setting x = −1 yields this for\n, when 0 ≤ k < n,\nand the general case follows by taking linear combinations of these.\nWhen P(x) is of degree less than or equal to n,\nwhere a_n is the coefficient of degree n in P(x).\nMore generally for ,\nwhere m and d are complex numbers.\nThis follows immediately applying to the polynomial Q(x):=P(m + dx) instead of P(x), and observing that Q(x) still has degree less than or equal to n, and that its coefficient of degree n is dnan.\nThe [series](https://bliptext.com/articles/series-mathematics) is convergent for k ≥ 2.\nThis formula is used in the analysis of the [German tank problem](https://bliptext.com/articles/german-tank-problem).\nIt follows from which is proved by [induction](https://bliptext.com/articles/mathematical-induction) on M.\n\n## Identities with combinatorial proofs\nMany identities involving binomial coefficients can be proved by [combinatorial means](https://bliptext.com/articles/combinatorial-proof).\nFor example, for nonnegative integers, the identity\n(which reduces to when q = 1) can be given a [double counting proof](https://bliptext.com/articles/double-counting-proof-technique), as follows.\nThe left side counts the number of ways of selecting a subset of [n] = {1, 2, ..., n} with at least q elements, and marking q elements among those selected.\nThe right side counts the same thing, because there are \\tbinom n q ways of choosing a set of q elements to mark, and 2^{n-q} to choose which of the remaining elements of [n] also belong to the subset.\nIn Pascal's identity\nboth sides count the number of k-element subsets of [n]: the two terms on the right side group them into those that contain element n and those that do not.\nThe identity also has a combinatorial proof.\nThe identity reads\nSuppose you have 2n empty squares arranged in a row and you want to mark (select) n of them.\nThere are ways to do this.\nOn the other hand, you may select your n squares by selecting k squares from among the first n and n-k squares from the remaining n squares; any k from 0 to n will work.\nThis gives\nNow apply to get the result.\nIf one denotes by\nF(i)\nthe sequence of [Fibonacci numbers](https://bliptext.com/articles/fibonacci-number), indexed so that\nF(0) = F(1) = 1\n, then the identity\nhas the following combinatorial proof.\nOne may show by [induction](https://bliptext.com/articles/mathematical-induction) that\nF(n)\ncounts the number of ways that a\nn × 1\nstrip of squares may be covered by\n2 × 1\nand\n1 × 1\ntiles.\nOn the other hand, if such a tiling uses exactly k of the\n2 × 1\ntiles, then it uses\nn − 2k\nof the\n1 × 1\ntiles, and so uses\nn − k\ntiles total.\nThere are ways to order these tiles, and so summing this coefficient over all possible values of k gives the identity.\n\n### Sum of coefficients row\nThe number of k-[combinations](https://bliptext.com/articles/combination) for all k,, is the sum of the nth row (counting from 0) of the binomial coefficients.\nThese combinations are enumerated by the 1 digits of the set of [base 2](https://bliptext.com/articles/base-2) numbers counting from 0 to 2^n - 1, where each digit position is an item from the set of n.\n\n## Dixon's identity\n[Dixon's identity](https://bliptext.com/articles/dixon-s-identity) is or, more generally,\nwhere a, b, and c are non-negative integers.\n\n## Continuous identities\nCertain trigonometric integrals have values expressible in terms of binomial coefficients: For any\nThese can be proved by using [Euler's formula](https://bliptext.com/articles/euler-s-formula) to convert [trigonometric functions](https://bliptext.com/articles/trigonometric-functions) to complex exponentials, expanding using the binomial theorem, and integrating term by term.\n\n## Congruences\nIf n is prime, then for every k with\nMore generally, this remains true if n is any number and k is such that all the numbers between 1 and k are coprime to n.\nIndeed, we have\n\n# Generating functions\n\n## Ordinary generating functions\nFor a fixed n, the [ordinary generating function](https://bliptext.com/articles/ordinary-generating-function) of the sequence is\nFor a fixed k, the ordinary generating function of the sequence is\nThe [bivariate generating function](https://bliptext.com/articles/bivariate-generating-function) of the binomial coefficients is\nA symmetric bivariate generating function of the binomial coefficients is which is the same as the previous generating function after the substitution x\\to xy.\n\n## Exponential generating function\nA symmetric [exponential bivariate generating function](https://bliptext.com/articles/exponential-generating-function) of the binomial coefficients is:\n\n# Divisibility properties\nIn 1852, [Kummer](https://bliptext.com/articles/ernst-kummer) proved that if m and n are nonnegative integers and p is a prime number, then the largest power of p dividing equals pc, where c is the number of carries when m and n are added in base p.\nEquivalently, the exponent of a prime p in \\tbinom n k equals the number of nonnegative integers j such that the [fractional part](https://bliptext.com/articles/fractional-part) of k/pj is greater than the [fractional part](https://bliptext.com/articles/fractional-part) of n/pj. It can be deduced from this that \\tbinom n k is divisible by n/[gcd](https://bliptext.com/articles/greatest-common-divisor)(n,k).\nIn particular therefore it follows that p divides for all positive integers r and s such that\ns < pr\n. However this is not true of higher powers of p: for example 9 does not divide.\nA somewhat surprising result by [David Singmaster](https://bliptext.com/articles/david-singmaster) (1974) is that any integer divides [almost all](https://bliptext.com/articles/almost-all) binomial coefficients.\nMore precisely, fix an integer d and let f(N) denote the number of binomial coefficients \\tbinom n k with n < N such that d divides \\tbinom n k. Then\nSince the number of binomial coefficients \\tbinom n k with n < N is N(N + 1) / 2, this implies that the density of binomial coefficients divisible by d goes to 1.\nBinomial coefficients have divisibility properties related to least common multiples of consecutive integers.\nFor example:\nAnother fact:\nAn integer\nn ≥ 2\nis prime if and only if all the intermediate binomial coefficients\nare divisible by n.\nProof:\nWhen p is prime, p divides\n0 < k < p\nbecause \\tbinom p k is a natural number and p divides the numerator but not the denominator.\nWhen n is composite, let p be the smallest prime factor of n and let\nk = n/p\n. Then\n0 < p < n\nand\notherwise the numerator\nk(n − 1)(n − 2)⋯(n − p + 1)\nhas to be divisible by\nn = k×p\n, this can only be the case when\n(n − 1)(n − 2)⋯(n − p + 1)\nis divisible by p.\nBut n is divisible by p, so p does not divide\nn − 1, n − 2, …, n − p + 1\nand because p is prime, we know that p does not divide\n(n − 1)(n − 2)⋯(n − p + 1)\nand so the numerator cannot be divisible by n.\n\n# Bounds and asymptotic formulas\nThe following bounds for \\tbinom n k hold for all values of n and k such that\n1 ≤ k ≤ n\nThe first inequality follows from the fact that\nand each of these k terms in this product is.\nA similar argument can be made to show the second inequality.\nThe final strict inequality is equivalent to, that is clear since the RHS is a term of the exponential series.\nFrom the divisibility properties we can infer that\nwhere both equalities can be achieved.\nThe following bounds are useful in information theory:\nwhere is the [binary entropy function](https://bliptext.com/articles/binary-entropy-function).\nIt can be further tightened to\nfor all.\n\n## Both n and k large\n[Stirling's approximation](https://bliptext.com/articles/stirling-s-approximation) yields the following approximation, valid when n-k,k both tend to infinity:\nBecause the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds.\nIn particular, when n is sufficiently large, one has\nand.\nMore generally, for\nm ≥ 2\nand\nn ≥ 1\n(again, by applying Stirling's formula to the factorials in the binomial coefficient),\nIf n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient.\nFor example, if then\nwhere d = n − 2k.\n\n## n much larger than k\nIf n is large and k is\no(n)\n(that is, if\nk/n → 0\n), then\nwhere again o is the [little o notation](https://bliptext.com/articles/little-o-notation).\n\n## Sums of binomial coefficients\nA simple and rough upper bound for the sum of binomial coefficients can be obtained using the [binomial theorem](https://bliptext.com/articles/binomial-theorem):\nMore precise bounds are given by\nvalid for all integers with.\n\n## Generalized binomial coefficients\nThe [infinite product formula for the gamma function](https://bliptext.com/articles/gamma-function) also gives an expression for binomial coefficients\nwhich yields the asymptotic formulas\nas.\nThis asymptotic behaviour is contained in the approximation\nas well.\n(Here H_k is the k-th [harmonic number](https://bliptext.com/articles/harmonic-number) and \\gamma is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant).)\nFurther, the asymptotic formula\nhold true, whenever k\\to\\infty and j/k \\to x for some complex number x.\n\n# Generalizations\n\n## Generalization to multinomials\nBinomial coefficients can be generalized to **multinomial coefficients** defined to be the number:\nwhere\nWhile the binomial coefficients represent the coefficients of\n(x + y)n\n, the multinomial coefficients\nrepresent the coefficients of the polynomial\nThe case r = 2 gives binomial coefficients:\nThe combinatorial interpretation of multinomial coefficients is distribution of n distinguishable elements over r (distinguishable) containers, each containing exactly ki elements, where i is the index of the container.\nMultinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation:\nand symmetry:\nwhere (\\sigma_i) is a [permutation](https://bliptext.com/articles/permutation) of (1, 2, ..., r).\n\n## Taylor series\nUsing [Stirling numbers of the first kind](https://bliptext.com/articles/stirling-numbers-of-the-first-kind) the [series expansion](https://bliptext.com/articles/taylor-series) around any arbitrarily chosen point z_0 is\n\n## Binomial coefficient with \n\nn = 1/2\nThe definition of the binomial coefficients can be extended to the case where n is real and k is integer.\nIn particular, the following identity holds for any non-negative integer k:\nThis shows up when expanding \\sqrt{1+x} into a power series using the Newton binomial series :\n\n## Products of binomial coefficients\nOne can express the product of two binomial coefficients as a linear combination of binomial coefficients:\nwhere the connection coefficients are [multinomial coefficients](https://bliptext.com/articles/multinomial-theorem).\nIn terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign\nm + n − k\nlabels to a pair of labelled combinatorial objects—of weight m and n respectively—that have had their first k labels identified, or glued together to get a new labelled combinatorial object of weight\nm + n − k\n. (That is, to separate the labels into three portions to apply to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what [falling factorials](https://bliptext.com/articles/falling-factorial) are to ordinary generating series.\nThe product of all binomial coefficients in the nth row of the Pascal triangle is given by the formula:\n\n## Partial fraction decomposition\nThe [partial fraction decomposition](https://bliptext.com/articles/partial-fraction-decomposition) of the reciprocal is given by\n\n## Newton's binomial series\nNewton's binomial series, named after [Sir Isaac Newton](https://bliptext.com/articles/sir-isaac-newton), is a generalization of the binomial theorem to infinite series:\nThe identity can be obtained by showing that both sides satisfy the [differential equation](https://bliptext.com/articles/differential-equation)\n(1 + z) f'(z) = α f(z)\n.\nThe [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of this series is 1.\nAn alternative expression is where the identity\nis applied.\n\n## Multiset (rising) binomial coefficient\nBinomial coefficients count subsets of prescribed size from a given set.\nA related combinatorial problem is to count [multisets](https://bliptext.com/articles/multiset) of prescribed size with elements drawn from a given set, that is, to count the number of ways to select a certain number of elements from a given set with the possibility of selecting the same element repeatedly.\nThe resulting numbers are called [multiset coefficients](https://bliptext.com/articles/multiset); the number of ways to \"multichoose\" (i.e., choose with replacement) k items from an n element set is denoted.\nTo avoid ambiguity and confusion with n's main denotation in this article,\nlet\nf = n = r + (k − 1)\nand\nr = f − (k − 1)\n.\nMultiset coefficients may be expressed in terms of binomial coefficients by the rule\nOne possible alternative characterization of this identity is as follows:\nWe may define the [falling factorial](https://bliptext.com/articles/falling-factorial) as\nand the corresponding rising factorial as\nso, for example,\nThen the binomial coefficients may be written as\nwhile the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:\n\n### Generalization to negative integers n\nFor any n,\nIn particular, binomial coefficients evaluated at negative integers n are given by signed multiset coefficients.\nIn the special case n = -1, this reduces to\nFor example, if n = −4 and k = 7, then r = 4 and f = 10:\n\n## Two real or complex valued arguments\nThe binomial coefficient is generalized to two real or complex valued arguments using the [gamma function](https://bliptext.com/articles/gamma-function) or [beta function](https://bliptext.com/articles/beta-function) via\nThis definition inherits these following additional properties from \\Gamma:\nmoreover,\nThe resulting function has been little-studied, apparently first being graphed in.\nNotably, many binomial identities fail: but for n positive (so -n negative).\nThe behavior is quite complex, and markedly different in various octants (that is, with respect to the x and y axes and the line y=x), with the behavior for negative x having singularities at negative integer values and a checkerboard of positive and negative regions:\n\n## Generalization to q-series\nThe binomial coefficient has a [q-analog](https://bliptext.com/articles/q-analog) generalization known as the [Gaussian binomial coefficient](https://bliptext.com/articles/gaussian-binomial-coefficient).\n\n## Generalization to infinite cardinals\nThe definition of the binomial coefficient can be generalized to [infinite cardinals](https://bliptext.com/articles/cardinal-number) by defining:\nwhere A is some set with [cardinality](https://bliptext.com/articles/cardinality) \\alpha.\nOne can show that the generalized binomial coefficient is well-defined, in the sense that no matter what set we choose to represent the [cardinal number](https://bliptext.com/articles/cardinal-number) \\alpha, will remain the same.\nFor finite cardinals, this definition coincides with the standard definition of the binomial coefficient.\nAssuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice), one can show that for any infinite cardinal \\alpha.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:30:12.661Z","updated_at":"2025-01-25T16:30:12.661Z","search_vector":"'binomi':1 'coeffici':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **binomial [coefficients](https://bliptext.com/articles/coefficient)** are the positive [integers](https://bliptext.com/articles/integer) that occur as [coefficients](https://bliptext.com/articles/coefficient) in the [binomial theorem](https://bliptext.com/articles/binomial-theorem).\nCommonly, a binomial coefficient is indexed by a pair of integers\nn ≥ k ≥ 0\nand is written It is the coefficient of the\nxk\nterm in the [polynomial expansion](https://bliptext.com/articles/polynomial-expansion) of the [binomial](https://bliptext.com/articles/binomial-polynomial) [power](https://bliptext.com/articles/exponentiation)\n(1 + x)n\nwhich using [factorial](https://bliptext.com/articles/factorial) notation can be compactly expressed as\nFor example, the fourth power of\n1 + x\nis and the binomial coefficient is the coefficient of the\nx2\nterm.\nArranging the numbers in successive rows for\nn = 0, 1, 2, ...\ngives a triangular array called [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle), satisfying the [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nThe binomial coefficients occur in many areas of mathematics, and especially in [combinatorics](https://bliptext.com/articles/combinatorics).\nIn combinatorics the symbol is usually read as \"n choose k\" because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements.\nFor example, there are ways to choose\n2\nelements from\n, namely\n,\n,\n,\n,\nand\n.\nThe first form of the binomial coefficients can be generalized to for any [complex number](https://bliptext.com/articles/complex-number) z and integer\nk ≥ 0\n, and many of their properties continue to hold in this more general form.\n# [The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. | upload.wikimedia.org/wikipedia/commons/f/f6/Pascal%27s///triangle///5.svg]\n# [Visualisation of binomial expansion up to the 4th power | upload.wikimedia.org/wikipedia/commons/7/7c/binomial///theorem///visualisation.svg]\n\n# History and notation\n[Andreas von Ettingshausen](https://bliptext.com/articles/andreas-von-ettingshausen) introduced the notation \\tbinom nk in 1826, although the numbers were known centuries earlier (see [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle)).\nIn about 1150, the Indian mathematician [Bhaskaracharya](https://bliptext.com/articles/bhaskaracharya) gave an exposition of binomial coefficients in his book [Līlāvatī](https://bliptext.com/articles/l-l-vat).\nAlternative notations include\nC(n, k)\n,\nnCk\n,\nnCk\n,\nCk n\n,\nCn k\n, and\nCn,k\n, in all of which the C stands for [combinations](https://bliptext.com/articles/combination) or choices; the C notation means the number of ways to choose k out of n objects.\nMany calculators use variants of the C notation because they can represent it on a single-line display.\nIn this form the binomial coefficients are easily compared to the numbers of [k-permutations of n](https://bliptext.com/articles/permutation), written as\nP(n, k)\n, etc.\n\n# Definition and interpretations\nFor [natural numbers](https://bliptext.com/articles/natural-number) (taken to include 0) n and k, the binomial [coefficient](https://bliptext.com/articles/coefficient) \\tbinom nk can be defined as the [coefficient](https://bliptext.com/articles/coefficient) of the [monomial](https://bliptext.com/articles/monomial)\nXk\nin the expansion of\n(1 + X)n\n. The same coefficient also occurs (if\nk ≤ n\n) in the [binomial formula](https://bliptext.com/articles/binomial-formula)\n(valid for any elements x, y of a [commutative ring](https://bliptext.com/articles/commutative-ring)),\nwhich explains the name \"binomial coefficient\".\nAnother occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-[combinations](https://bliptext.com/articles/combination)) of an n-element set.\nThis number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power\n(1 + X)n\none temporarily labels the term X with an index i (running from\n1\nto n), then each subset of k indices gives after expansion a contribution\nXk\n, and the coefficient of that monomial in the result will be the number of such subsets.\nThis shows in particular that \\tbinom nk is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n [bits](https://bliptext.com/articles/bit) (digits 0 or 1) whose sum is k is given by \\tbinom nk, while the number of ways to write where every\nai\nis a nonnegative integer is given by \\tbinom{n+k-1}{n-1}.\nMost of these interpretations can be shown to be equivalent to counting k-combinations.\n\n# Computing the value of binomial coefficients\nSeveral methods exist to compute the value of without actually expanding a binomial power or counting k-combinations.\n\n## Recursive formula\nOne method uses the [recursive](https://bliptext.com/articles/recursion), purely additive formula\nfor all integers n,k such that\nwith boundary values\nfor all integers\nn ≥ 0\n.\nThe formula follows from considering the set\nand counting separately (a) the k-element groupings that include a particular set element, say \"i\", in every group (since \"i\" is already chosen to fill one spot in every group, we need only choose\nk − 1\nfrom the remaining\nn − 1\n) and (b) all the k-groupings that don't include \"i\"; this enumerates all the possible k-combinations of n elements. It also follows from tracing the contributions to Xk in\n(1 + X)n−1(1 + X)\n. As there is zero\nXn+1\nor\nX−1\nin\n(1 + X)n\n, one might extend the definition beyond the above boundaries to include when either\nk > n\nor\nk 0\n. This latter result is also a special case of the result from the theory of [finite differences](https://bliptext.com/articles/finite-differences) that for any polynomial P(x) of degree less than n,\nDifferentiating k times and setting x = −1 yields this for\n, when 0 ≤ k < n,\nand the general case follows by taking linear combinations of these.\nWhen P(x) is of degree less than or equal to n,\nwhere a_n is the coefficient of degree n in P(x).\nMore generally for ,\nwhere m and d are complex numbers.\nThis follows immediately applying to the polynomial Q(x):=P(m + dx) instead of P(x), and observing that Q(x) still has degree less than or equal to n, and that its coefficient of degree n is dnan.\nThe [series](https://bliptext.com/articles/series-mathematics) is convergent for k ≥ 2.\nThis formula is used in the analysis of the [German tank problem](https://bliptext.com/articles/german-tank-problem).\nIt follows from which is proved by [induction](https://bliptext.com/articles/mathematical-induction) on M.\n\n## Identities with combinatorial proofs\nMany identities involving binomial coefficients can be proved by [combinatorial means](https://bliptext.com/articles/combinatorial-proof).\nFor example, for nonnegative integers, the identity\n(which reduces to when q = 1) can be given a [double counting proof](https://bliptext.com/articles/double-counting-proof-technique), as follows.\nThe left side counts the number of ways of selecting a subset of [n] = {1, 2, ..., n} with at least q elements, and marking q elements among those selected.\nThe right side counts the same thing, because there are \\tbinom n q ways of choosing a set of q elements to mark, and 2^{n-q} to choose which of the remaining elements of [n] also belong to the subset.\nIn Pascal's identity\nboth sides count the number of k-element subsets of [n]: the two terms on the right side group them into those that contain element n and those that do not.\nThe identity also has a combinatorial proof.\nThe identity reads\nSuppose you have 2n empty squares arranged in a row and you want to mark (select) n of them.\nThere are ways to do this.\nOn the other hand, you may select your n squares by selecting k squares from among the first n and n-k squares from the remaining n squares; any k from 0 to n will work.\nThis gives\nNow apply to get the result.\nIf one denotes by\nF(i)\nthe sequence of [Fibonacci numbers](https://bliptext.com/articles/fibonacci-number), indexed so that\nF(0) = F(1) = 1\n, then the identity\nhas the following combinatorial proof.\nOne may show by [induction](https://bliptext.com/articles/mathematical-induction) that\nF(n)\ncounts the number of ways that a\nn × 1\nstrip of squares may be covered by\n2 × 1\nand\n1 × 1\ntiles.\nOn the other hand, if such a tiling uses exactly k of the\n2 × 1\ntiles, then it uses\nn − 2k\nof the\n1 × 1\ntiles, and so uses\nn − k\ntiles total.\nThere are ways to order these tiles, and so summing this coefficient over all possible values of k gives the identity.\n\n### Sum of coefficients row\nThe number of k-[combinations](https://bliptext.com/articles/combination) for all k,, is the sum of the nth row (counting from 0) of the binomial coefficients.\nThese combinations are enumerated by the 1 digits of the set of [base 2](https://bliptext.com/articles/base-2) numbers counting from 0 to 2^n - 1, where each digit position is an item from the set of n.\n\n## Dixon's identity\n[Dixon's identity](https://bliptext.com/articles/dixon-s-identity) is or, more generally,\nwhere a, b, and c are non-negative integers.\n\n## Continuous identities\nCertain trigonometric integrals have values expressible in terms of binomial coefficients: For any\nThese can be proved by using [Euler's formula](https://bliptext.com/articles/euler-s-formula) to convert [trigonometric functions](https://bliptext.com/articles/trigonometric-functions) to complex exponentials, expanding using the binomial theorem, and integrating term by term.\n\n## Congruences\nIf n is prime, then for every k with\nMore generally, this remains true if n is any number and k is such that all the numbers between 1 and k are coprime to n.\nIndeed, we have\n\n# Generating functions\n\n## Ordinary generating functions\nFor a fixed n, the [ordinary generating function](https://bliptext.com/articles/ordinary-generating-function) of the sequence is\nFor a fixed k, the ordinary generating function of the sequence is\nThe [bivariate generating function](https://bliptext.com/articles/bivariate-generating-function) of the binomial coefficients is\nA symmetric bivariate generating function of the binomial coefficients is which is the same as the previous generating function after the substitution x\\to xy.\n\n## Exponential generating function\nA symmetric [exponential bivariate generating function](https://bliptext.com/articles/exponential-generating-function) of the binomial coefficients is:\n\n# Divisibility properties\nIn 1852, [Kummer](https://bliptext.com/articles/ernst-kummer) proved that if m and n are nonnegative integers and p is a prime number, then the largest power of p dividing equals pc, where c is the number of carries when m and n are added in base p.\nEquivalently, the exponent of a prime p in \\tbinom n k equals the number of nonnegative integers j such that the [fractional part](https://bliptext.com/articles/fractional-part) of k/pj is greater than the [fractional part](https://bliptext.com/articles/fractional-part) of n/pj. It can be deduced from this that \\tbinom n k is divisible by n/[gcd](https://bliptext.com/articles/greatest-common-divisor)(n,k).\nIn particular therefore it follows that p divides for all positive integers r and s such that\ns < pr\n. However this is not true of higher powers of p: for example 9 does not divide.\nA somewhat surprising result by [David Singmaster](https://bliptext.com/articles/david-singmaster) (1974) is that any integer divides [almost all](https://bliptext.com/articles/almost-all) binomial coefficients.\nMore precisely, fix an integer d and let f(N) denote the number of binomial coefficients \\tbinom n k with n < N such that d divides \\tbinom n k. Then\nSince the number of binomial coefficients \\tbinom n k with n < N is N(N + 1) / 2, this implies that the density of binomial coefficients divisible by d goes to 1.\nBinomial coefficients have divisibility properties related to least common multiples of consecutive integers.\nFor example:\nAnother fact:\nAn integer\nn ≥ 2\nis prime if and only if all the intermediate binomial coefficients\nare divisible by n.\nProof:\nWhen p is prime, p divides\n0 < k < p\nbecause \\tbinom p k is a natural number and p divides the numerator but not the denominator.\nWhen n is composite, let p be the smallest prime factor of n and let\nk = n/p\n. Then\n0 < p < n\nand\notherwise the numerator\nk(n − 1)(n − 2)⋯(n − p + 1)\nhas to be divisible by\nn = k×p\n, this can only be the case when\n(n − 1)(n − 2)⋯(n − p + 1)\nis divisible by p.\nBut n is divisible by p, so p does not divide\nn − 1, n − 2, …, n − p + 1\nand because p is prime, we know that p does not divide\n(n − 1)(n − 2)⋯(n − p + 1)\nand so the numerator cannot be divisible by n.\n\n# Bounds and asymptotic formulas\nThe following bounds for \\tbinom n k hold for all values of n and k such that\n1 ≤ k ≤ n\nThe first inequality follows from the fact that\nand each of these k terms in this product is.\nA similar argument can be made to show the second inequality.\nThe final strict inequality is equivalent to, that is clear since the RHS is a term of the exponential series.\nFrom the divisibility properties we can infer that\nwhere both equalities can be achieved.\nThe following bounds are useful in information theory:\nwhere is the [binary entropy function](https://bliptext.com/articles/binary-entropy-function).\nIt can be further tightened to\nfor all.\n\n## Both n and k large\n[Stirling's approximation](https://bliptext.com/articles/stirling-s-approximation) yields the following approximation, valid when n-k,k both tend to infinity:\nBecause the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds.\nIn particular, when n is sufficiently large, one has\nand.\nMore generally, for\nm ≥ 2\nand\nn ≥ 1\n(again, by applying Stirling's formula to the factorials in the binomial coefficient),\nIf n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient.\nFor example, if then\nwhere d = n − 2k.\n\n## n much larger than k\nIf n is large and k is\no(n)\n(that is, if\nk/n → 0\n), then\nwhere again o is the [little o notation](https://bliptext.com/articles/little-o-notation).\n\n## Sums of binomial coefficients\nA simple and rough upper bound for the sum of binomial coefficients can be obtained using the [binomial theorem](https://bliptext.com/articles/binomial-theorem):\nMore precise bounds are given by\nvalid for all integers with.\n\n## Generalized binomial coefficients\nThe [infinite product formula for the gamma function](https://bliptext.com/articles/gamma-function) also gives an expression for binomial coefficients\nwhich yields the asymptotic formulas\nas.\nThis asymptotic behaviour is contained in the approximation\nas well.\n(Here H_k is the k-th [harmonic number](https://bliptext.com/articles/harmonic-number) and \\gamma is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant).)\nFurther, the asymptotic formula\nhold true, whenever k\\to\\infty and j/k \\to x for some complex number x.\n\n# Generalizations\n\n## Generalization to multinomials\nBinomial coefficients can be generalized to **multinomial coefficients** defined to be the number:\nwhere\nWhile the binomial coefficients represent the coefficients of\n(x + y)n\n, the multinomial coefficients\nrepresent the coefficients of the polynomial\nThe case r = 2 gives binomial coefficients:\nThe combinatorial interpretation of multinomial coefficients is distribution of n distinguishable elements over r (distinguishable) containers, each containing exactly ki elements, where i is the index of the container.\nMultinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation:\nand symmetry:\nwhere (\\sigma_i) is a [permutation](https://bliptext.com/articles/permutation) of (1, 2, ..., r).\n\n## Taylor series\nUsing [Stirling numbers of the first kind](https://bliptext.com/articles/stirling-numbers-of-the-first-kind) the [series expansion](https://bliptext.com/articles/taylor-series) around any arbitrarily chosen point z_0 is\n\n## Binomial coefficient with \n\nn = 1/2\nThe definition of the binomial coefficients can be extended to the case where n is real and k is integer.\nIn particular, the following identity holds for any non-negative integer k:\nThis shows up when expanding \\sqrt{1+x} into a power series using the Newton binomial series :\n\n## Products of binomial coefficients\nOne can express the product of two binomial coefficients as a linear combination of binomial coefficients:\nwhere the connection coefficients are [multinomial coefficients](https://bliptext.com/articles/multinomial-theorem).\nIn terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign\nm + n − k\nlabels to a pair of labelled combinatorial objects—of weight m and n respectively—that have had their first k labels identified, or glued together to get a new labelled combinatorial object of weight\nm + n − k\n. (That is, to separate the labels into three portions to apply to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what [falling factorials](https://bliptext.com/articles/falling-factorial) are to ordinary generating series.\nThe product of all binomial coefficients in the nth row of the Pascal triangle is given by the formula:\n\n## Partial fraction decomposition\nThe [partial fraction decomposition](https://bliptext.com/articles/partial-fraction-decomposition) of the reciprocal is given by\n\n## Newton's binomial series\nNewton's binomial series, named after [Sir Isaac Newton](https://bliptext.com/articles/sir-isaac-newton), is a generalization of the binomial theorem to infinite series:\nThe identity can be obtained by showing that both sides satisfy the [differential equation](https://bliptext.com/articles/differential-equation)\n(1 + z) f'(z) = α f(z)\n.\nThe [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of this series is 1.\nAn alternative expression is where the identity\nis applied.\n\n## Multiset (rising) binomial coefficient\nBinomial coefficients count subsets of prescribed size from a given set.\nA related combinatorial problem is to count [multisets](https://bliptext.com/articles/multiset) of prescribed size with elements drawn from a given set, that is, to count the number of ways to select a certain number of elements from a given set with the possibility of selecting the same element repeatedly.\nThe resulting numbers are called [multiset coefficients](https://bliptext.com/articles/multiset); the number of ways to \"multichoose\" (i.e., choose with replacement) k items from an n element set is denoted.\nTo avoid ambiguity and confusion with n's main denotation in this article,\nlet\nf = n = r + (k − 1)\nand\nr = f − (k − 1)\n.\nMultiset coefficients may be expressed in terms of binomial coefficients by the rule\nOne possible alternative characterization of this identity is as follows:\nWe may define the [falling factorial](https://bliptext.com/articles/falling-factorial) as\nand the corresponding rising factorial as\nso, for example,\nThen the binomial coefficients may be written as\nwhile the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:\n\n### Generalization to negative integers n\nFor any n,\nIn particular, binomial coefficients evaluated at negative integers n are given by signed multiset coefficients.\nIn the special case n = -1, this reduces to\nFor example, if n = −4 and k = 7, then r = 4 and f = 10:\n\n## Two real or complex valued arguments\nThe binomial coefficient is generalized to two real or complex valued arguments using the [gamma function](https://bliptext.com/articles/gamma-function) or [beta function](https://bliptext.com/articles/beta-function) via\nThis definition inherits these following additional properties from \\Gamma:\nmoreover,\nThe resulting function has been little-studied, apparently first being graphed in.\nNotably, many binomial identities fail: but for n positive (so -n negative).\nThe behavior is quite complex, and markedly different in various octants (that is, with respect to the x and y axes and the line y=x), with the behavior for negative x having singularities at negative integer values and a checkerboard of positive and negative regions:\n\n## Generalization to q-series\nThe binomial coefficient has a [q-analog](https://bliptext.com/articles/q-analog) generalization known as the [Gaussian binomial coefficient](https://bliptext.com/articles/gaussian-binomial-coefficient).\n\n## Generalization to infinite cardinals\nThe definition of the binomial coefficient can be generalized to [infinite cardinals](https://bliptext.com/articles/cardinal-number) by defining:\nwhere A is some set with [cardinality](https://bliptext.com/articles/cardinality) \\alpha.\nOne can show that the generalized binomial coefficient is well-defined, in the sense that no matter what set we choose to represent the [cardinal number](https://bliptext.com/articles/cardinal-number) \\alpha, will remain the same.\nFor finite cardinals, this definition coincides with the standard definition of the binomial coefficient.\nAssuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice), one can show that for any infinite cardinal \\alpha.","revisionCount":0},"hasMore":false,"nextCursor":null}