{{short description|Number of subsets of a given size}}{{redirect|nCk||NCK (disambiguation)}}Image:Pascal's triangle 5.svg|right|thumb|200px|The binomial coefficients can be arranged to form Pascal's trianglePascal's trianglethumb|300px|Visualisation of binomial expansion up to the 4th powerIn mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers {{math|n ≥ k ≥ 0}} and is written tbinom{n}{k}. It is the coefficient of the {{math|xk}} term in the polynomial expansion of the binomial power {{math|(1 + x)n}}; this coefficient can be computed by the multiplicative formula which using factorial notation can be compactly expressed as For example, the fourth power of {{math|1 + x}} is (1 + x)^4 &= tbinom{4}{0} x^0 + tbinom{4}{1} x^1 + tbinom{4}{2} x^2 + tbinom{4}{3} x^3 + tbinom{4}{4} x^4 &= 1 + 4x + 6 x^2 + 4x^3 + x^4,end{align}and the binomial coefficient tbinom{4}{2} =tfrac{4times 3}{2times1} = tfrac{4!}{2!2!} = 6 is the coefficient of the {{math|x2}} term.Arranging the numbers tbinom{n}{0}, tbinom{n}{1}, ldots, tbinom{n}{n} in successive rows for {{math|1= n = 0, 1, 2, ...}} gives a triangular array called Pascal's triangle, satisfying the recurrence relation The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol tbinom{n}{k} is usually read as "{{mvar|n}} choose {{mvar|k}}" because there are tbinom{n}{k} ways to choose an (unordered) subset of {{mvar|k}} elements from a fixed set of {{mvar|n}} elements. For example, there are tbinom{4}{2}=6 ways to choose {{math|2}} elements from {{math|{{mset|1, 2, 3, 4}}}}, namely {{math|{{mset|1, 2}}}}, {{math|{{mset|1, 3}}}}, {{math|{{mset|1, 4}}}}, {{math|{{mset|2, 3}}}}, {{math|{{mset|2, 4}}}} and {{math|{{mset|3, 4}}}}.The binomial coefficients can be generalized to tbinom{z}{k} for any complex number {{mvar|z}} and integer {{math|k ≥ 0}}, and many of their properties continue to hold in this more general form.

History and notation

Andreas von Ettingshausen introduced the notation tbinom nk in 1826,{{harvtxt|Higham|1998}} although the numbers were known centuries earlier (see Pascal's triangle). In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī.Lilavati Section 6, Chapter 4 (see {{harvtxt|Knuth|1997}}).Alternative notations include {{math|C(n, k)}}, {{math|{{sub|n}}C{{sub|k}}}}, {{math|{{sup|n}}Ck}}, {{math|C{{su|p=k|b=n|lh=0.8}}}},{{harvnb|Uspensky|1937|p=18}} {{math|C{{su|p=n|b=k|lh=0.8}}}}, and {{math|C{{sub|n,k}}}}, in all of which the {{mvar|C}} stands for combinations or choices. Many calculators use variants of the {{nowrap|{{mvar|C}} notation}} because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to {{mvar|k}}-permutations of {{mvar|n}}, written as {{math|P(n, k)}}, etc.">

Definition and interpretations {| class"wikitable" style"float:right;line-spacing:0.8;margin-left:1ex;"

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The first few binomial coefficientson a left-aligned Pascal's triangle

For natural numbers (taken to include 0) n and k, the binomial coefficient tbinom nk can be defined as the coefficient of the monomial Xk in the expansion of {{math|(1 + X)n}}. The same coefficient also occurs (if {{math|k ≤ n}}) in the binomial formula{{NumBlk|:|(x+y)^n=sum_{k=0}^nbinom nk x^ky^{n-k}|{{EquationRef|∗}}}}(valid for any elements x, y of a commutative ring),which explains the name "binomial coefficient".Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that {{mvar|k}} objects can be chosen from among {{mvar|n}} objects; more formally, the number of {{mvar|k}}-element subsets (or {{mvar|k}}-combinations) of an {{mvar|n}}-element set. This 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 {{mvar|n}} factors of the power {{math|(1 + X)n}} one temporarily labels the term {{mvar|X}} with an index {{mvar|i}} (running from {{math|1}} to {{mvar|n}}), then each subset of {{mvar|k}} indices gives after expansion a contribution {{math|X'k}}, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that tbinom nk is a natural number for any natural numbers {{mvar|n}} and {{mvar|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 {{mvar|n}} bits (digits 0 or 1) whose sum is {{mvar|k}} is given by tbinom nk, while the number of ways to write k = a_1 + a_2 + cdots + a_n where every {{math|a'i}} is a nonnegative integer is given by {{tmath|1= tbinom{n+k-1}{n-1} }}. Most of these interpretations can be shown to be equivalent to counting {{mvar|k}}-combinations.

Computing the value of binomial coefficients

Several methods exist to compute the value of tbinom{n}{k} without actually expanding a binomial power or counting {{mvar|k}}-combinations.

Recursive formula

One method uses the recursive, purely additive formula with boundary valuesbinom n0 = binom nn = 1for all integers {{math|1=n ≥ 0}}.The formula follows from considering the set {{math|{{mset|1, 2, 3, ..., n}}}} and counting separately (a) the {{mvar|k}}-element groupings that include a particular set element, say "{{mvar|i}}", in every group (since "{{mvar|i}}" is already chosen to fill one spot in every group, we need only choose {{math|k − 1}} from the remaining {{math|n − 1}}) and (b) all the k-groupings that don't include "{{mvar|i}}"; this enumerates all the possible {{mvar|k}}-combinations of {{mvar|n}} elements. It also follows from tracing the contributions to X'k in {{math|(1 + X)n−1(1 + X)}}. As there is zero {{math|X'n+1}} or {{math|X−1}} in {{math|(1 + X)n}}, one might extend the definition beyond the above boundaries to include tbinom nk = 0 when either {{math|k > n}} or {{math|k < 0}}. This recursive formula then allows the construction of Pascal's triangle, surrounded by white spaces where the zeros, or the trivial coefficients, would be.

Multiplicative formula

A more efficient method to compute individual binomial coefficients is given by the formulabinom nk = frac{n^{underline{k}}}{k!} = frac{n(n-1)(n-2)cdots(n-(k-1))}{k(k-1)(k-2)cdots 1} = prod_{i=1}^kfrac{ n+1-i}{ i},where the numerator of the first fraction n^{underline{k}} is expressed as a falling factorial power.This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.The numerator gives the number of ways to select a sequence of {{mvar|k}} distinct objects, retaining the order of selection, from a set of {{mvar|n}} objects. The denominator counts the number of distinct sequences that define the same {{mvar|k}}-combination when order is disregarded.Due to the symmetry of the binomial coefficient with regard to {{mvar|k}} and {{math|n − k}}, calculation may be optimised by setting the upper limit of the product above to the smaller of {{mvar|k}} and {{math|n − k}}.

Factorial formula

Finally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar factorial function: where {{math|n!}} denotes the factorial of {{mvar|n}}. This formula follows from the multiplicative formula above by multiplying numerator and denominator by {{math|(n − k)!}}; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation (in the case that {{mvar|k}} is small and {{mvar|n}} is large) unless common factors are first cancelled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions){{NumBlk|:| binom nk = binom n{n-k} quad text{for } 0leq kleq n,|{{EquationRef|1}}}}which leads to a more efficient multiplicative computational routine. Using the falling factorial notation, begin{cases} end{cases}.

Generalization and connection to the binomial series

The multiplicative formula allows the definition of binomial coefficients to be extendedSee {{Harv|Graham|Knuth|Patashnik|1994}}, which also defines tbinom n k = 0 for kRow number {{mvar|n}} contains the numbers 0}}. This latter result is also a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,JOURNAL, Ruiz, Sebastian, An Algebraic Identity Leading to Wilson's Theorem, The Mathematical Gazette, 1996, 80, 489, 579–582, 10.2307/3618534, 3618534, math/0406086, 125556648, Differentiating ({{EquationNote|2}}) k times and setting x = −1 yields this forP(x)=x(x-1)cdots(x-k+1),when 0 ≤ k < n,and the general case follows by taking linear combinations of these.When P(x) is of degree less than or equal to n,{{NumBlk|:| sum_{j=0}^n (-1)^jbinom n j P(n-j) = n!a_n|{{EquationRef|10}}}}where a_n is the coefficient of degree n in P(x).More generally for ({{EquationNote|10}}), where m and d are complex numbers. This follows immediately applying ({{EquationNote|10}}) to the polynomial {{tmath|1=Q(x):=P(m + dx)}} instead of {{tmath|P(x)}}, and observing that {{tmath|Q(x)}} still has degree less than or equal to n, and that its coefficient of degree n is dnan.The series frac{k-1}{k} sum_{j=0}^infty frac 1 {binom {j+x} k}= frac 1 {binom{x-1}{k-1}} is convergent for k ≥ 2. This formula is used in the analysis of the German tank problem. It follows from frac{k-1}ksum_{j=0}^{M}frac 1 {binom{j+x} k}=frac 1{binom{x-1}{k-1}}-frac 1{binom{M+x}{k-1}} which is proved by induction on M.

Identities with combinatorial proofs

Many identities involving binomial coefficients can be proved by combinatorial means. For example, for nonnegative integers {n} geq {q}, the identity (which reduces to ({{EquationNote|6}}) when q = 1) can be given a double counting proof, as follows. The 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. The 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.In Pascal's identity both 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.The identity ({{EquationNote|8}}) also has a combinatorial proof. The identity reads Suppose you have 2n empty squares arranged in a row and you want to mark (select) n of them. There are tbinom {2n}n ways to do this. On 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. This gives Now apply ({{EquationNote|1}}) to get the result.If one denotes by {{math|F(i)}} the sequence of Fibonacci numbers, indexed so that {{math|1=F(0) = F(1) = 1}}, then the identitysum_{k=0}^{leftlfloorfrac{n}{2}rightrfloor} binom {n-k} k = F(n)has the following combinatorial proof.{{harvnb|Benjamin|Quinn|2003|loc=pp. 4−5}} One may show by induction that {{math|F(n)}} counts the number of ways that a {{math|n × 1}} strip of squares may be covered by {{math|2 × 1}} and {{math|1 × 1}} tiles. On the other hand, if such a tiling uses exactly {{mvar|k}} of the {{math|2 × 1}} tiles, then it uses {{math|n − 2k}} of the {{math|1 × 1}} tiles, and so uses {{math|n − k}} tiles total. There are tbinom{n-k}{k} ways to order these tiles, and so summing this coefficient over all possible values of {{mvar|k}} gives the identity.

Sum of coefficients row

{{See also|Combination#Number of k-combinations for all k}}The number of k-combinations for all k, sum_{0leq{k}leq{n}}binom nk = 2^n, is the sum of the nth row (counting from 0) of the binomial coefficients. These combinations are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2^n - 1, where each digit position is an item from the set of n.

Dixon's identity

Dixon's identity is or, more generally, where a, b, and c are non-negative integers.

Continuous identities

Certain trigonometric integrals have values expressible in terms of binomial coefficients: For any m, n in N, int_{-pi}^{pi} sin((2m-n)x)sin^n(x) dx =begin{cases}(-1)^{m+(n+1)/2} frac{pi}{2^{n-1}} binom{n}{m}, & n text{ odd} end{cases} int_{-pi}^{pi} cos((2m-n)x)sin^n(x) dx =begin{cases}(-1)^{m+(n/2)} frac{pi}{2^{n-1}} binom{n}{m}, & n text{ even} end{cases}These can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.

Congruences

If n is prime, then binom {n-1}k equiv (-1)^k mod n for every k with 0leq k leq n-1.More 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.Indeed, we have binom {n-1} k = {(n-1)(n-2)cdots(n-k)over 1cdot 2cdots k}

prod_{i1}^{k}{n-iover i}equiv prod_{i1}^{k}{-iover i} (-1)^k mod n.

Generating functions

Ordinary generating functions

For a fixed {{mvar|n}}, the ordinary generating function of the sequence tbinom n0,tbinom n1,tbinom n2,ldots is For a fixed {{mvar|k}}, the ordinary generating function of the sequence tbinom 0k,tbinom 1k, tbinom 2k,ldots, is The bivariate generating function of the binomial coefficients is A symmetric bivariate generating function of the binomial coefficients is which is the same as the previous generating function after the substitution xto xy.

Exponential generating function

A symmetric exponential bivariate generating function of the binomial coefficients is:

Divisibility properties

In 1852, Kummer proved that if m and n are nonnegative integers and p is a prime number, then the largest power of p dividing tbinom{m+n}{m} equals pc, where c is the number of carries when m and n are added in base p.Equivalently, the exponent of a prime p in tbinom n kequals the number of nonnegative integers j such that the fractional part of k/p'j is greater than the fractional part of n/p'j. It can be deduced from this that tbinom n k is divisible by n/gcd(n,k). In particular therefore it follows that p divides tbinom{p^r}{s} for all positive integers r and s such that {{math|s < pr}}. However this is not true of higher powers of p: for example 9 does not divide tbinom{9}{6}.A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More 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 Since 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.Binomial coefficients have divisibility properties related to least common multiples of consecutive integers. For example:JOURNAL, Farhi, Bakir, Nontrivial lower bounds for the least common multiple of some finite sequence of integers, Journal of Number Theory, 125, 2, 2007, 393–411, 10.1016/j.jnt.2006.10.017, 0803.0290, 115167580, Another fact:An integer {{math|n ≥ 2}} is prime if and only ifall the intermediate binomial coefficients are divisible by n.Proof:When p is prime, p divides because tbinom p k is a natural number and p divides the numerator but not the denominator.When n is composite, let p be the smallest prime factor of n and let {{math|1=k = n/p}}. Then {{math|1=0 < p < n}} and otherwise the numerator {{math|1=k(n − 1)(n − 2)⋯(n − p + 1)}} has to be divisible by {{math|1=n = k×p}}, this can only be the case when {{math|1=(n − 1)(n − 2)⋯(n − p + 1)}} is divisible by p. But n is divisible by p, so p does not divide {{math|1=n − 1, n − 2, …, n − p + 1}} and because p is prime, we know that p does not divide {{math|1=(n − 1)(n − 2)⋯(n − p + 1)}} and so the numerator cannot be divisible by n.

Bounds and asymptotic formulas

The following bounds for tbinom n k hold for all values of n and k such that {{math|1 ≤ k ≤ n}}:frac{n^k}{k^k} le {n choose k} le frac{n^k}{k!} < left(frac{ncdot e}{k}right)^k.The first inequality follows from the fact that and each of these k terms in this product is geq frac{n}{k} . A similar argument can be made to show the second inequality. The final strict inequality is equivalent to e^k > k^k / k!, that is clear since the RHS is a term of the exponential series e^k = sum_{j=0}^infty k^j/j! .From the divisibility properties we can infer thatfrac{operatorname{lcm}(n-k, ldots, n)}{(n-k) cdot operatorname{lcm}left(binom{k}{0}, ldots, binom{k}{k}right)}leqbinom{n}{k} leq frac{operatorname{lcm}(n-k, ldots, n)}{n - k},where both equalities can be achieved.The following bounds are useful in information theory:BOOK, Thomas M. Cover, Joy A. Thomas, Elements of Information Theory, 18 July 2006, Wiley, Hoboken, New Jersey, 0-471-24195-4, {{rp|353}} where H(p) = -plog_2(p) -(1-p)log_2(1-p) is the binary entropy function. It can be further tightened to for all 1 leq k leq n-1.BOOK, F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, 16, 3rd, 1981, North-Holland, 0-444-85009-0, {{rp|309}}

Both n and k large

Stirling's approximation yields the following approximation, valid when n-k,k both tend to infinity:{n choose k} simsqrt{nover 2pi k (n-k)} cdot{n^n over k^k (n-k)^{n-k}}Because the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds.In particular, when n is sufficiently large, one has sqrt{n}{mn choose n} ge frac{m^{m(n-1)+1}}{(m-1)^{(m-1)(n-1)}}.If n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient binom{n}{k}. For example, if | n/2 - k | = o(n^{2/3}) then where d = n − 2k.BOOK, Asymptopia, Spencer, Joel, Florescu, Laura, 2014, American Mathematical Society, AMS, 978-1-4704-0904-3, Student mathematical library, 71, 66, 865574788, Joel Spencer,

{{mvar|n}} much larger than {{mvar|k}}

If {{mvar|n}} is large and {{mvar|k}} is {{math|o(n)}} (that is, if {{math| k/n → 0}}), then where again {{mvar|o}} is the little o notation.BOOK, Asymptopia, Spencer, Joel, Florescu, Laura, 2014, American Mathematical Society, AMS, 978-1-4704-0904-3, Student mathematical library, 71, 59, 865574788, Joel Spencer,

Sums of binomial coefficients

A simple and rough upper bound for the sum of binomial coefficients can be obtained using the binomial theorem:sum_{i=0}^k {n choose i} leq sum_{i=0}^k n^icdot 1^{k-i} leq (1+n)^kMore precise bounds are given byfrac{1}{sqrt{8nvarepsilon(1-varepsilon)}} cdot 2^{H(varepsilon) cdot n} leq sum_{i=0}^{k} binom{n}{i} leq 2^{H(varepsilon) cdot n},valid for all integers n > k geq 1 with varepsilon doteq k/n leq 1/2.see e.g. {{harvtxt|Ash|1990|p=121}} or {{harvtxt|Flum|Grohe|2006|p=427}}.

Generalized binomial coefficients

The infinite product formula for the gamma function also gives an expression for binomial coefficients(-1)^k {z choose k}= {-z+k-1 choose k} = frac{1}{Gamma(-z)} frac{1}{(k+1)^{z+1}} prod_{j=k+1} frac{left(1+frac{1}{j}right)^{-z-1}}{1-frac{z+1}{j}}which yields the asymptotic formulas{z choose k} approx frac{(-1)^k}{Gamma(-z) k^{z+1}} qquad text{and} qquad {z+k choose k} = frac{k^z}{Gamma(z+1)}left( 1+frac{z(z+1)}{2k}+mathcal{O}left(k^{-2}right)right)as k to infty.This asymptotic behaviour is contained in the approximation{z+k choose k}approx frac{e^{z(H_k-gamma)}}{Gamma(z+1)}as well. (Here H_k is the k-th harmonic number and gamma is the Euler–Mascheroni constant.)Further, the asymptotic formulafrac{{z+kchoose j}}{{kchoose j}}to left(1-frac{j}{k}right)^{-z}quadtext{and}quad frac{{jchoose j-k}}{{j-zchoose j-k}}to left(frac{j}{k}right)^zhold true, whenever ktoinfty and j/k to x for some complex number x.

Generalizations

Generalization to multinomials

Binomial coefficients can be generalized to multinomial coefficients defined to be the number: where While the binomial coefficients represent the coefficients of {{math|(x + y)n}}, the multinomial coefficientsrepresent the coefficients of the polynomial The case r = 2 gives binomial coefficients: The 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.Multinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation: and symmetry: where (sigma_i) is a permutation of (1, 2, ..., r).

Taylor series

Using Stirling numbers of the first kind the series expansion around any arbitrarily chosen point z_0 is

Binomial coefficient with {{math|1n 1/2}}

The definition of the binomial coefficients can be extended to the case where n is real and k is integer.In particular, the following identity holds for any non-negative integer k: This shows up when expanding sqrt{1+x} into a power series using the Newton binomial series :

Products of binomial coefficients

One can express the product of two binomial coefficients as a linear combination of binomial coefficients: where the connection coefficients are multinomial coefficients. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign {{math|m + n − k}} labels 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 {{math|m + n − k}}. (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 are to ordinary generating series.The product of all binomial coefficients in the nth row of the Pascal triangle is given by the formula:

Partial fraction decomposition

The partial fraction decomposition of the reciprocal is given by qquad frac{1}{{z+n choose n}}= sum_{i=1}^n (-1)^{i-1} {n choose i} frac{i}{z+i}.

Newton's binomial series

Newton's binomial series, named after Sir Isaac Newton, is a generalization of the binomial theorem to infinite series: The identity can be obtained by showing that both sides satisfy the differential equation {{math|1=(1 + z) f'(z) = α f(z)}}.The radius of convergence of this series is 1. An alternative expression is where the identity is applied.

Multiset (rising) binomial coefficient

Binomial coefficients count subsets of prescribed size from a given set. A related combinatorial problem is to count multisets 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. The resulting numbers are called multiset coefficients;{{citation }}. the number of ways to "multichoose" (i.e., choose with replacement) k items from an n element set is denoted left(!!binom n k!!right).To avoid ambiguity and confusion with n's main denotation in this article, let {{math|1=f = n = r + (k − 1)}} and {{math|1=r = f − (k − 1)}}.Multiset coefficients may be expressed in terms of binomial coefficients by the rulebinom{f}{k}=left(!!binom{r}{k}!!right)=binom{r+k-1}{k}.One possible alternative characterization of this identity is as follows:We may define the falling factorial as(f)_{k}=f^{underline k}=(f-k+1)cdots(f-3)cdot(f-2)cdot(f-1)cdot f,and the corresponding rising factorial asr^{(k)}=,r^{overline k}=,rcdot(r+1)cdot(r+2)cdot(r+3)cdots(r+k-1);so, for example,17cdot18cdot19cdot20cdot21=(21)_{5}=21^{underline 5}=17^{overline 5}=17^{(5)}.Then the binomial coefficients may be written asbinom{f}{k} = frac{(f)_{k}}{k!} =frac{(f-k+1)cdots(f-2)cdot(f-1)cdot f}{1cdot2cdot3cdot4cdot5cdots k} ,while the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:left(!!binom{r}{k}!!right)=frac{r^{(k)}}{k!}=frac{rcdot(r+1)cdot(r+2)cdots(r+k-1)}{1cdot2cdot3cdot4cdot5cdots k}.

Generalization to negative integers n

{{Pascal_triangle_extended.svg}}For any n, &=(-1)^k;frac{ncdot(n+1)cdot(n+2)cdots (n + k - 1)}{k!}&=(-1)^kbinom{n + k - 1}{k}&=(-1)^kleft(!!binom{n}{k}!!right);.end{align}In particular, binomial coefficients evaluated at negative integers n are given by signed multiset coefficients. In the special case n = -1, this reduces to (-1)^k=binom{-1}{k}=left(!!binom{-k}{k}!!right) .For example, if n = −4 and k = 7, then r = 4 and f = 10: {-10cdot-9cdot-8cdot-7cdot-6cdot-5cdot-4}{1cdot2cdot3cdot4cdot5cdot6cdot7}&=(-1)^7;frac{4cdot5cdot6cdot7cdot8cdot9cdot10}{1cdot2cdot3cdot4cdot5cdot6cdot7}&=left(!!binom{-7}{7}!!right)left(!!binom{4}{7}!!right)=binom{-1}{7}binom{10}{7}.end{align}

Two real or complex valued arguments

The binomial coefficient is generalized to two real or complex valued arguments using the gamma function or beta function via This definition inherits these following additional properties from Gamma: moreover, The resulting function has been little-studied, apparently first being graphed in {{Harv|Fowler|1996}}. Notably, many binomial identities fail: binom{n }{ m} = binom{n }{ n-m} but binom{-n}{m} neq binom{-n}{-n-m} for n positive (so -n negative). The 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:

• in the octant 0 leq y leq x it is a smoothly interpolated form of the usual binomial, with a ridge ("Pascal's ridge").
• in the octant 0 leq x leq y and in the quadrant x geq 0, y leq 0 the function is close to zero.
• in the quadrant x leq 0, y geq 0 the function is alternatingly very large positive and negative on the parallelograms with vertices (-n,m+1), (-n,m), (-n-1,m-1), (-n-1,m)
• in the octant 0 > x > y the behavior is again alternatingly very large positive and negative, but on a square grid.
• in the octant -1 > y > x + 1 it is close to zero, except for near the singularities.

Generalization to q-series

The binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.

Generalization to infinite cardinals

The definition of the binomial coefficient can be generalized to infinite cardinals by defining: where {{mvar|A}} is some set with cardinality alpha. One 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 alpha, {alpha choose beta} will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.Assuming the Axiom of Choice, one can show that {alpha choose alpha} = 2^{alpha} for any infinite cardinal alpha.

See also

{{div col|colwidth=25em}}

• Binomial transform
• Delannoy number
• Eulerian number
• Hypergeometric function
• List of factorial and binomial topics
• Macaulay representation of an integer
• Motzkin number
• Multiplicities of entries in Pascal's triangle
• Narayana number
• Star of David theorem
• Sun's curious identity
• Table of Newtonian series
• Trinomial expansion

{{div col end}}

Notes

{{reflist|25em}}

References

• BOOK, Robert B., Ash, Information Theory, Dover Publications, Inc., 1965, 1990, 0-486-66521-6,
• BOOK

• BOOK, Victor, Bryant, Victor Bryant, Aspects of combinatorics, Cambridge University Press, 1993, 0-521-41974-3,
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• JOURNAL, 10.2307/2975209, The Binomial Coefficient Function, David, Fowler, David Fowler (mathematician), The American Mathematical Monthly, 103, January 1996, 1–17, 1, Mathematical Association of America, 2975209,
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• BOOK, Ronald L., Graham, Ronald Graham, Donald E., Knuth, Donald Knuth, Oren, Patashnik, Oren Patashnik, Concrete Mathematics,weblink limited, Addison-Wesley, 1994, Second, 0-201-55802-5, 153–256,
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• BOOK, Donald E., Knuth, Donald Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third, Addison-Wesley, 1997, 0-201-89683-4, 52–74,
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External links

• {{springer|title=Binomial coefficients|id=p/b016410}}
• JOURNAL, Andrew Granville,weblink Arithmetic Properties of Binomial Coefficients I. Binomial coefficients modulo prime powers, CMS Conf. Proc, 20, 151–162, 1997, Andrew Granville, 2013-09-03,weblink" title="web.archive.org/web/20150923201436weblink">weblink 2015-09-23, dead,

{{PlanetMath attribution title = Binomial Coefficient title2 = Upper and lower bounds to binomial coefficient title3 = Binomial coefficient is an integer title4 = Generalized binomial coefficients}}