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Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \( \tbinom nk \) , and it is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n. Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.
This family of numbers also arises in many other areas than algebra, notably in combinatorics. For any set containing n elements, the number of distinct k-element subsets of it that can be formed (the k-combinations of its elements) is given by the binomial coefficient \tbinom nk. Therefore \tbinom nk is often read as "n choose k". The properties of binomial coefficients have led to extending the meaning of the symbol \tbinom nk beyond the basic case where n and k are nonnegative integers with k ≤ n; such expressions are then still called binomial coefficients.
The notation \tbinom nk was introduced by Andreas von Ettingshausen in 1826,[1] although the numbers were already known centuries before that (see Pascal's triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pingala's chandaḥśāstra. In about 1150, the Hindu mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book Lilavati.[2]
Alternative notations include C(n, k), nCk, nCk, \( \scriptstyle C^{k}_{n}, \scriptstyle C^{n}_{k} \),[3] in all of which the C stands for combinations or choices.
Definition and interpretations
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 (1 + X)n. The same coefficient also occurs (if k ≤ n) in the binomial formula
\( (x+y)^n=\sum_{k=0}^n\binom nk x^{n-k}y^k \)
(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 k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an 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 n factors of the power (1 + X)n one temporarily labels the term X with an index i (running from 1 to n), then each subset of k indices gives after expansion a contribution Xk, 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 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 (digits 0 or 1) whose sum is k is given by \( \tbinom nk \), while the number of ways to write \( k=a_1+a_2+\cdots+a_n \)where every ai is a nonnegative integer is given by \tbinom{n+k-1}k. Most of these interpretations are easily seen to be equivalent to counting k-combinations.
Computing the value of binomial coefficients
Several methods exist to compute the value of \( \tbinom nk \) without actually expanding a binomial power or counting k-combinations.
Recursive formula
One has a recursive formula for binomial coefficients
\( \binom nk = \binom{n-1}{k-1} + \binom{n-1}k \quad \mbox{for all integers }n,k>0, \)
with initial values
\( \binom n0 = 1 \quad \mbox{for all integers } n\ge0, \)
\( \binom 0k = 0 \quad \mbox{for all integers } k>0. \)
The formula follows either from tracing the contributions to Xk in (1 + X)n−1(1 + X), or by counting k-combinations of {1, 2, ..., n} that contain n and that do not contain n separately. It follows easily that \( \tbinom nk=0 \) when k > n, and \( \tbinom nn=1 \) for all n, so the recursion can stop when reaching such cases. This recursive formula then allows the construction of Pascal's triangle.
Multiplicative formula
A more efficient method to compute individual binomial coefficients is given by the formula
\( \binom nk = \frac{n^{\underline{k}}}{k!} = \frac{n(n-1)(n-2)\cdots(n-(k-1))}{k(k-1)(k-2)\cdots 1}=\prod_{i=1}^k \frac{n-(k-i)}{i}, \)
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 k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded.
Factorial formula
Finally there is a formula using factorials that is easy to remember:
\( \binom nk = \frac{n!}{k!\,(n-k)!} \quad \mbox{for }\ 0\leq k\leq n. \)
where n! denotes the factorial of n. This formula follows from the multiplicative formula above by multiplying numerator and denominator by (n − k)!; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation unless common factors are first canceled (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)
\( \binom nk = \binom n{n-k} \quad \mbox{for }\ 0\leq k\leq n. \)
(1)
Generalization and connection to the binomial series
The multiplicative formula allows the definition of binomial coefficients to be extended[4] by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible:
\( \binom \alpha k = \frac{\alpha^{\underline k}}{k!} = \frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k(k-1)(k-2)\cdots 1} \quad\mbox{for } k \in N \mbox{ and arbitrary } \alpha. \)
With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the \tbinom\alpha k binomial coefficients:
\( (1+X)^\alpha = \sum_{k=0}^\infty {\alpha \choose k} X^k. \)
(2)
This formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an identity of formal power series in X, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation, notably
\( (1+X)^\alpha(1+X)^\beta=(1+X)^{\alpha+\beta} \quad\mbox{and}\quad ((1+X)^\alpha)^\beta=(1+X)^{\alpha\beta}. \)
If α is a nonnegative integer n, then all terms with k > n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However for other values of α, including negative integers and rational numbers, the series is really infinite.
Pascal's triangle
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.
Main article: Pascal's rule
Main article: Pascal's triangle
Pascal's rule is the important recurrence relation
\( {n \choose k} + {n \choose k+1} = {n+1 \choose k+1}, \)
(3)
which can be used to prove by mathematical induction that \tbinom n k is a natural number for all n and k, (equivalent to the statement that k! divides the product of k consecutive integers), a fact that is not immediately obvious from formula (1).
Pascal's rule also gives rise to Pascal's triangle:
0: 1
1: 1 1
2: 1 2 1
3: 1 3 3 1
4: 1 4 6 4 1
5: 1 5 10 10 5 1
6: 1 6 15 20 15 6 1
7: 1 7 21 35 35 21 7 1
8: 1 8 28 56 70 56 28 8 1
Row number n contains the numbers \tbinom n k for k = 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that
(x + y)5 = 1 x5 + 5 x4y + 10 x3y2 + 10 x2y3 + 5 x y3 + 1 y5.
The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation (3) above.
Combinatorics and statistics
Binomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:
There are \( \tbinom n k \)ways to choose k elements from a set of n elements. See Combination.
There are \( \tbinom {n+k-1}k \)ways to choose k elements from a set of n if repetitions are allowed. See Multiset.
There are \( \tbinom {n+k} k \) strings containing k ones and n zeros.
There are \( \tbinom {n+1} k \) strings consisting of k ones and n zeros such that no two ones are adjacent.[5]
The Catalan numbers are \( \frac {\tbinom{2n}n}{n+1}. \)
The binomial distribution in statistics is \( \tbinom n k p^k (1-p)^{n-k} \!. \)
The formula for a Bézier curve.
Binomial coefficients as polynomials
For any nonnegative integer k, the expression \( \scriptstyle{\binom{t}{k}} \) can be simplified and defined as a polynomial divided by k!:
\( \binom{t}{k} =\frac{(t)_k}{k!}=\frac{(t)_k}{(k)_k}= \frac{t(t-1)(t-2)\cdots(t-k+1)}{k(k-1)(k-2)\cdots(2)(1)};\,\! \)
This presents a polynomial in t with rational coefficients.
As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem.
For each k, the polynomial \( \tbinom{t}{k} \) can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) = ... = p(k − 1) = 0 and p(k) = 1.
Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:
\( \binom{t}{k} = \sum_{i=0}^k \frac{s_{k,i}}{k!} t^i. \)
The derivative of \tbinom{t}{k} can be calculated by logarithmic differentiation:
\( \frac{\mathrm{d}}{\mathrm{d}t} \binom{t}{k} = \binom{t}{k} \sum_{i=0}^{k-1} \frac{1}{t-i}\,. \)
Binomial coefficients as a basis for the space of polynomials
Over any field containing Q, each polynomial p(t) of degree at most d is uniquely expressible as a linear combination \( \sum_{k=0}^d a_k \binom{t}{k} \). The coefficient ak is the kth difference of the sequence p(0), p(1), …, p(k). Explicitly,[6]
\( a_k = \sum_{i=0}^k (-1)^{k-i} \binom{k}{i} p(i). \)
(3.5)
Integer-valued polynomials
Each polynomial \( \tbinom{t}{k} \) is integer-valued: it takes integer values at integer inputs. (One way to prove this is by induction on k, using Pascal's identity.) Therefore any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (3.5) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More 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.
Example
The integer-valued polynomial 3t(3t + 1)/2 can be rewritten as
\( 9\tbinom{t}{2} + 6 \tbinom{t}{1} + 0\tbinom{t}{0}.\ \)
Identities involving binomial coefficients
The factorial formula facilitates relating nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, then
\( \binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1} \)
(4)
and, with a little more work,
\( \binom {n-1}{k} - \binom{n-1}{k-1} = \frac{n-2k}{n} \binom{n}{k}. \)
Moreover, the following may be useful:
\( \binom{N}{h}\binom{N-h}{k}=\binom{N}{k}\binom{N-k}{h} \)
Series involving binomial coefficients
The formula
\( \sum_{k=0}^n \tbinom n k = 2^n \)
(5)
is obtained from (2) using x = 1. This is equivalent to saying that the elements in one row of Pascal's triangle always add up to two raised to an integer power. A combinatorial interpretation of this fact involving double counting is given by counting subsets of size 0, size 1, size 2, and so on up to size n of a set S of n elements. Since we count the number of subsets of size i for 0 ≤ i ≤ n, this sum must be equal to the number of subsets of S, which is known to be 2n. That is, Equation 5 is a statement that the power set for a finite set with n elements has size 2n. More explicitly, consider a bit string with n digits. This bit string can be used to represent 2n numbers. Now consider all of the bit strings with no ones in them. There is just one, or rather n choose 0. Next consider the number of bit strings with just a single one in them. There are n, or rather n choose 1. Continuing this way we can see that the equation above holds.
The formulas
\( \sum_{k=0}^n k \tbinom n k = n 2^{n-1} \)
(6a)
and
\( \sum_{k=0}^n k^2 \tbinom n k = (n + n^2)2^{n-2} \)
(6b)
follow from (2), after differentiating with respect to x (twice in the latter) and then substituting x = 1.
The Chu-Vandermonde identity, which holds for any complex-values m and n and any non-negative integer k, is
\( \sum_{j=0}^k \tbinom m j \tbinom{n-m}{k-j} = \tbinom n k \)
(7a)
and can be found by examination of the coefficient of \( x^k \) in the expansion of (1 + x)m (1 + x)n − m = (1 + x)n using equation (2). When m = 1, equation (7a) reduces to equation (3).
A similar looking formula, which applies for any integers j, k, and n satisfying 0 ≤ j ≤ k ≤ n, is
\( \sum_{m=0}^n \tbinom m j \tbinom {n-m}{k-j}= \tbinom {n+1}{k+1}\,, \)
(7b)
and can be found by examination of the coefficient of x^{n+1} in the expansion of \( x\left(\tfrac{x^{j}}{(1-x)^{j+1}}\right) \left(\tfrac{x^{k-j}}{(1-x)^{k-j+1}}\right) = \tfrac{x^{k+1}}{(1-x)^{k+2}} \) using \( \tfrac{x^l}{(1-x)^{l+1}} = \sum_{p=0}^\infty \tbinom p l x^p\, \). When j = k, equation (7b) gives
\( \sum_{m=0}^n \tbinom m k = \tbinom {n+1}{k+1}\,. \)
From expansion (7a) using n = 2m, k = m, and (1), one finds
\( \sum_{j=0}^m \tbinom m j ^2 = \tbinom {2m} m. \)
(8)
Let F(n) denote the n-th Fibonacci number. We obtain a formula about the diagonals of Pascal's triangle
\( \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \tbinom {n-k} k = F(n+1). \)
(9)
This can be proved by induction using (3) or by Zeckendorf's representation (Just note that the lhs gives the number of subsets of {F(2),...,F(n)} without consecutive members, which also form all the numbers below F(n+1)).
Also using (3) and induction, one can show that
\( \sum_{j=k}^n \tbinom j k = \tbinom {n+1}{k+1}. \)
(10)
Although there is no closed formula for
\( \sum_{j=0}^k \tbinom n j \)
(unless one resorts to Hypergeometric functions), one can again use (3) and induction, to show that for k = 0, ..., n−1
\( \sum_{j=0}^k (-1)^j\tbinom n j = (-1)^k\tbinom {n-1}k \)
(11)
as well as
\( \sum_{j=0}^n (-1)^j\tbinom n j = 0 \)
(12)
[except in the trivial case where n = 0, where the result is 1 instead] which is itself a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,
\( \sum_{j=0}^n (-1)^j\tbinom n j P(j) = 0. \)
(13a)
Differentiating (2) k times and setting x = −1 yields this for \( P(x)=x(x-1)\cdots(x-k+1) \), 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,
\( \sum_{j=0}^n (-1)^j\tbinom n j P(n-j) = n!a_n \)
(13b)
where a_n is the coefficient of degree n in P(x).
More generally for (13b),
\( \sum_{j=0}^n (-1)^j\tbinom n j P(m+(n-j)d) = d^n n! a_n \)
(13c)
where m and d are complex numbers. This follows immediately applying (13b) to the polynomial Q(x):=P(m + dx) instead of P(x), and observing that Q(x) has still degree less than or equal to n, and that its coefficient of degree n is dnan.
The infinite series
\( \sum_{j=m}^\infty \frac 1 {\binom j k}=\frac k{(k-1)\binom{m-1}{k-1}} \)
(14)
is convergent for k ≥ 2. This formula is used in the analysis of the German tank problem. It is equivalent to the formula for the finite sum
\( \sum_{j=m}^{M-1}\frac 1 {k\binom j k}=\frac 1{(k-1)\binom{m-1}{k-1}}-\frac 1{(k-1)\binom{M-1}{k-1}} \)
which is proved for M>m by induction on M.
Using (8) one can derive
\( \sum_{i=0}^{n}{i\binom{n}{i}^2}=\frac{n}{2}\binom{2n}{n} \)
(15)
and
\( \sum_{i=0}^n{i^2\binom{n}{i}^2}=n^2 \binom{2n-2}{n-1}. \)
(16)
Series multisection gives the following identity for the sum of binomial coefficients taken with a step s and offset t (0\leqslant t<s) as a closed-form sum of s terms:
\( \binom{n}{t}+\binom{n}{t+s}+\binom{n}{t+2s}+\ldots=\frac{1}{s}\sum_{j=0}^{s-1}\left(2\cos\frac{\pi j}{s}\right)^n\cos\frac{\pi(n-2t)j}{s}. \)
Identities with combinatorial proofs
Many identities involving binomial coefficients can be proved by combinatorial means. For example, the following identity for nonnegative integers \( {n} \geq {q} \) (which reduces to (6a) when q = 1):
\( \sum_{k=q}^n \tbinom n k \tbinom k q = 2^{n-q}\tbinom n q \)
(16b)
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 k elements, and marking q elements among those selected. The right side counts the same parameter, because there are \tbinom n q ways of choosing a set of q marks and they occur in all subsets that additionally contain some subset of the remaining elements, of which there are \( 2^{n-q} \).
In the Pascal's rule
\( {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k} \)
both sides count the number of k-element subsets of [n] with the right hand side first grouping them into those that contain element n and those that do not.
The identity (8) also has a combinatorial proof. The identity reads
\( \sum_{k=0}^n \tbinom n k ^2 = \tbinom {2n} n. \)
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 1 to n will work. This gives
\( \sum_{k=0}^n\tbinom n k\tbinom n{n-k} = \tbinom {2n} n. \)
Now apply (4) to get the result.
Sum of coefficients row
See also: Combination#Number of k-combinations for all k
The number of k-combinations for all k, \( \sum_{0\leq{k}\leq{n}}\binom nk = 2^n, \) 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
\( \sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^3 =\frac{(3a)!}{(a!)^3} \)
or, more generally,
\( \sum_{k=-a}^a(-1)^k{a+b\choose a+k} {b+c\choose b+k}{c+a\choose c+k} = \frac{(a+b+c)!}{a!\,b!\,c!}\,, \)
where a, b, and c are non-negative integers.
Continuous identities
Certain trigonometric integrals have values expressible in terms of binomial coefficients:
For \( \textstyle m, n \in \mathbb{Z} \) and \( \textstyle m, n \geq 0 \)
\( \int_{-\pi}^{\pi} \cos((2m-n)x)\cos^n x\ dx = \frac{\pi}{2^{n-1}} \binom{n}{m} \)
(17)
\( \int_{-\pi}^{\pi} \sin((2m-n)x)\sin^n x\ dx = \left \{ \begin{array}{cc} (-1)^{m+(n+1)/2} \frac{\pi}{2^{n-1}} \binom{n}{m} & n \mbox{ odd} \\ 0 & \mbox{otherwise} \\ \end{array} \right. \)
(18)
\( \int_{-\pi}^{\pi} \cos((2m-n)x)\sin^n x\ dx = \left \{ \begin{array}{cc} (-1)^{m+(n+1)/2} \frac{\pi}{2^{n-1}} \binom{n}{m} & n \mbox{ even} \\ 0 & \mbox{otherwise} \\ \end{array} \right. \)
(19)
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.
Generating functions
Ordinary generating functions
For a fixed n, the ordinary generating function of the sequence \( {n\choose 0},\;{n\choose 1},\;{n\choose 2},\;\ldots \) is:
\( \sum_k {n\choose k} x^k = (1+x)^n. \)
For a fixed k, the ordinary generating function of the sequence \({0\choose k},\;{1\choose k},\;{2\choose k},\;\ldots \) is:
\( \sum_n {n\choose k} y^n = \frac{y^k}{(1-y)^{k+1}}. \)
The bivariate generating function of the binomial coefficients is:
\( \sum_{n,k} {n\choose k} x^k y^n = \frac{1}{1-y-xy}. \)
Another bivariate generating function of the binomial coefficients, which is symmetric, is:
\( \sum_{n,k} {n+k\choose k} x^k y^n = \frac{1}{1-x-y}. \)
Exponential generating function
The exponential bivariate generating function of the binomial coefficients is:
\( \sum_{n,k} \frac{1}{(n+k)!}{n+k\choose k} x^k y^n = e^{x+y} \).
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 \( \tbinom n k \) equals the number of nonnegative integers j such that the fractional part of k/pj is greater than the fractional part of n/pj. It can be deduced from this that \( \tbinom n k \) is divisible by n/gcd(n,k).
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
\( \lim_{N\to\infty} \frac{f(N)}{N(N+1)/2} = 1. \)
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.
Another fact: An integer n ≥ 2 is prime if and only if all the intermediate binomial coefficients
\( \binom n 1, \binom n 2, \ldots, \binom n{n-1} \)
are divisible by n.
Proof: When p is prime, p divides
\( \binom p k = \frac{p \cdot (p-1) \cdots (p-k+1)}{k \cdot (k-1) \cdots 1} \) for all 0 < k < p
because it is a natural number and the numerator has a prime factor p but the denominator does not have a prime factor p.
When n is composite, let p be the smallest prime factor of n and let k = n/p. Then 0 < p < n and
\( \binom n p = \frac{n(n-1)(n-2)\cdot\ldots\cdot(n-p+1)}{p!}=\frac{k(n-1)(n-2)\cdot\ldots\cdot(n-p+1)}{(p-1)!}\not\equiv 0 \pmod{n} \)
otherwise the numerator k(n−1)(n−2)×...×(n−p+1) has to be divisible by n = k×p, this can only be the case when (n−1)(n−2)×...×(n−p+1) is divisible by p. But n is divisible by p, so p does not divide n−1, n−2, ..., n−p+1 and because p is prime, we know that p does not divide (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:
\( \left(\frac{n}{k}\right)^k \le {n \choose k} \le \frac{n^k}{k!} \le \left(\frac{n\cdot e}{k}\right)^k. \)
Stirling's approximation yields the bounds:
\( \sqrt{n}{2n \choose n} \ge 2^{2n-1} \) and, in general, \( \sqrt{n}{mn \choose n} \ge \frac{m^{m(n-1)+1}}{(m-1)^{(m-1)(n-1)}} \) for m ≥ 2 and n ≥ 1,
and the approximation
\( {2n \choose n} \sim \frac{4^n}{\sqrt{\pi n}} as n\rightarrow\infty\,. \)
The infinite product formula (cf. Gamma function, alternative definition)
\( (-1)^k {z \choose k}= {-z+k-1 \choose k} = \frac{1}{\Gamma(-z)} \frac{1}{(k+1)^{z+1}} \prod_{j=k+1} \frac{(1+\frac{1}{j})^{-z-1}}{1-\frac{z+1}{j}} \)
yields the asymptotic formulas
\( {z \choose k} \approx \frac{(-1)^k}{\Gamma(-z) k^{z+1}} \qquad \mathrm{and} \qquad {z+k \choose k} = \frac{k^z}{\Gamma(z+1)}\left( 1+\frac{z(z+1)}{2k}+\mathcal{O}\left(k^{-2}\right)\right) \)
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).
The sum of binomial coefficients can be bounded by a term exponential in n and the binary entropy of the largest n/k that occurs. More precisely, for \( n\geq 1 \) and \( 0<\epsilon<\frac{1}{2} \), it holds
\( \sum_{k=0}^{\lfloor\epsilon n\rfloor} {n \choose k} \leq 2^{H(\epsilon) \cdot n}, \)
where \( H(\epsilon)=-\epsilon\cdot \log \epsilon - (1-\epsilon)\cdot \log (1-\epsilon) \) is the binary entropy of \( \epsilon \).[7]
A simple and rough upper bound for the sum of binomial coefficients is given by the formula below (not difficult to prove)
\( \sum_{i=0}^k {n \choose i} \leq (n+1)^k \)
Generalizations
Generalization to multinomials
Binomial coefficients can be generalized to multinomial coefficients. They are defined to be the number:
\( {n\choose k_1,k_2,\ldots,k_r} =\frac{n!}{k_1!k_2!\cdots k_r!} \)
where
\( \sum_{i=1}^rk_i=n. \)
While the binomial coefficients represent the coefficients of (x+y)n, the multinomial coefficients represent the coefficients of the polynomial
\( (x_1 + x_2 + \cdots + x_r)^n.\ \)
See multinomial theorem. The case r = 2 gives binomial coefficients:
\( {n\choose k_1,k_2}={n\choose k_1, n-k_1}={n\choose k_1}= {n\choose k_2}. \)
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 these of binomial coefficients, for example the recurrence relation:
\( {n\choose k_1,k_2,\ldots,k_r} ={n-1\choose k_1-1,k_2,\ldots,k_r}+{n-1\choose k_1,k_2-1,\ldots,k_r}+\ldots+{n-1\choose k_1,k_2,\ldots,k_r-1} \)
and symmetry:
\( {n\choose k_1,k_2,\ldots,k_r} ={n\choose k_{\sigma_1},k_{\sigma_2},\ldots,k_{\sigma_r}} \)
where \( (\sigma_i) \( is a permutation of (1,2,...,r).
Generalization to negative integers
If \( k \geq 0 \), then \( {n \choose k} = \frac{n(n-1) \dots (n-k+1)}{1 \cdot 2 \cdots k}= (-1)^k {-n+k-1 \choose k} \) extends to all n .
In the special case n = -1, this reduces to \( \binom{-1}{k} = (-1)^k\,. \)
Taylor series
Using Stirling numbers of the first kind the series expansion around any arbitrarily chosen point z_0 is
\( \begin{align} {z \choose k} = \frac{1}{k!}\sum_{i=0}^k z^i s_{k,i}&=\sum_{i=0}^k (z- z_0)^i \sum_{j=i}^k {z_0 \choose j-i} \frac{s_{k+i-j,i}}{(k+i-j)!} \\ &=\sum_{i=0}^k (z-z_0)^i \sum_{j=i}^k z_0^{j-i} {j \choose i} \frac{s_{k,j}}{k!}.\end{align} \)
Binomial coefficient with n=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 :
\( {{1/2}\choose{k}}={{2k}\choose{k}}\frac{(-1)^{k+1}}{2^{2k}(2k-1)} \)
This shows up when expanding \sqrt{1+x} into a power series using the Newton binomial series :
\( \sqrt{1+x}=\sum_{k\geqslant0}{\tbinom{1/2}{k}}x^k \)
Identity for the product of binomial coefficients
One can express the product of binomial coefficients as a linear combination of binomial coefficients:
\( {z \choose m} {z\choose n} = \sum_{k=0}^m {m+n-k\choose k,m-k,n-k} {z\choose m+n-k} \)
where the connection coefficients are multinomial coefficients. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign 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 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.
Partial Fraction Decomposition
The partial fraction decomposition of the inverse is given by
\( \frac{1}{{z \choose n}}= \sum_{i=0}^{n-1} (-1)^{n-1-i} {n \choose i} \frac{n-i}{z-i}, and \frac{1}{{z+n \choose n}}= \sum_{i=1}^n (-1)^{i-1} {n \choose i} \frac{i}{z+i}. \)
Newton's binomial series
Newton's binomial series, named after Sir Isaac Newton, is one of the simplest Newton series:
\( (1+z)^{\alpha} = \sum_{n=0}^{\infty}{\alpha\choose n}z^n = 1+{\alpha\choose1}z+{\alpha\choose 2}z^2+\cdots. \)
The identity can be obtained by showing that both sides satisfy the differential equation (1+z) f'(z) = α f(z).
The radius of convergence of this series is 1. An alternative expression is
\( \frac{1}{(1-z)^{\alpha+1}} = \sum_{n=0}^{\infty}{n+\alpha \choose n}z^n \)
where the identity
\( {n \choose k} = (-1)^k {k-n-1 \choose k} \)
is applied.
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
\( {x \choose y}= \frac{\Gamma(x+1)}{\Gamma(y+1) \Gamma(x-y+1)}= \frac{1}{(x+1) B(x-y+1,y+1)}. \)
This definition inherits these following additional properties from \Gamma:
\( {x \choose y}= \frac{\sin (y \pi)}{\sin(x \pi)} {-y-1 \choose -x-1}= \frac{\sin((x-y) \pi)}{\sin (x \pi)} {y-x-1 \choose y}; \)
moreover,
\( {x \choose y} \cdot {y \choose x}= \frac{\sin((x-y) \pi)}{(x-y) \pi}. \)
The resulting function has been little-studied, apparently first being graphed in (Fowler 1996). Notably, many binomial identities fail: \( \textstyle{{n \choose m} = {n \choose n-m}} \) but \( \textstyle{{-n \choose m} = {-n \choose -n-m}} \) 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:
\( {\alpha \choose \beta} = | \{ B \subseteq A : |B| = \beta \} | \)
where 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 \( \alpha. \)
Binomial coefficient in programming languages
The notation \( {n \choose k} \) is convenient in handwriting but inconvenient for typewriters and computer terminals. Many programming languages do not offer a standard subroutine for computing the binomial coefficient, but for example the J programming language uses the exclamation mark: k ! n .
Naive implementations of the factorial formula, such as the following snippet in Python:
def binomialCoefficient(n, k):
from math import factorial
return factorial(n) // (factorial(k) * factorial(n - k))
are very slow and are uselessly calculating factorials of very high numbers (in languages as C or Java they suffer from overflow errors because of this reason). A direct implementation of the multiplicative formula works well:
def binomialCoefficient(n, k):
if k < 0 or k > n:
return 0
if k > n - k: # take advantage of symmetry
k = n - k
c = 1
for i in range(k):
c = c * (n - (k - (i+1)))
c = c // (i+1)
return c
(Notice that range(k) is a list from 0 to k-1 and, as a consequence, we need to use i+1 in the above function). The example mentioned above can be also written in functional style. The following Scheme example uses recursive definition
\( {n \choose k+1} = \frac{n-k}{k+1} {n \choose k} \)
Rational arithmetic can be easily avoided using integer division
\( {n \choose k+1} = \left[(n-k) {n \choose k}\right] \div (k+1) \)
The following implementation uses all these ideas
(define (binomial n k)
;; Helper function to compute C(n,k) via forward recursion
(define (binomial-iter n k i prev)
(if (>= i k)
prev
(binomial-iter n k (+ i 1) (/ (* (- n i) prev) (+ i 1)))))
;; Use symmetry property C(n,k)=C(n, n-k)
(if (< k (- n k))
(binomial-iter n k 0 1)
(binomial-iter n (- n k) 0 1)))
Another way to compute the binomial coefficient when using large numbers is to recognize that
\( {n \choose k} = \frac{n!}{k!\,(n-k)!} = \frac{\Gamma(n+1)}{\Gamma(k+1)\,\Gamma(n-k+1)} = \exp(\ln\Gamma(n+1)-\ln\Gamma(k+1)-\ln\Gamma(n-k+1)), \)
where \( \ln \Gamma(n) \) denotes the natural logarithm of the gamma function at n. It is a special function that is easily computed and is standard in some programming languages such as using log_gamma in Maxima, LogGamma in Mathematica, or gammaln in MATLAB. Roundoff error may cause the returned value to not be an integer.
See also
Central binomial coefficient
Binomial transform
Star of David theorem
Table of Newtonian series
List of factorial and binomial topics
Multiplicities of entries in Pascal's triangle
Sun's curious identity
Kummer's theorem on prime-power divisors of binomial coefficients
Notes
^ Higham (1998)
^ Lilavati Section 6, Chapter 4 (see Knuth (1997)).
^ Shilov (1977)
^ See (Graham, Knuth & Patashnik 1994), which also defines \tbinom n k = 0 for k<0. Alternative generalizations, such as to two real or complex valued arguments using the Gamma function assign nonzero values to \tbinom n k for k<0, but this causes most binomial coefficient identities to fail, and thus is not widely used majority of definitions. One such choice of nonzero values leads to the aesthetically pleasing "Pascal windmill" in Hilton, Holton and Pedersen, Mathematical reflections: in a room with many mirrors, Springer, 1997, but causes even Pascal's identity to fail (at the origin).
^ Muir, Thomas (1902). "Note on Selected Combinations". Proceedings of the Royal Society of Edinburgh.
^ This can be seen as a discrete analog of Taylor's theorem. It is closely related to Newton's polynomial. Alternating sums of this form may be expressed as the Nörlund–Rice integral.
^ see e.g. Flum & Grohe (2006, p. 427)
References
Benjamin, Arthur T.; Quinn, Jennifer (2003). Proofs that Really Count: The Art of Combinatorial Proof , Mathematical Association of America.
Bryant, Victor (1993). Aspects of combinatorics. Cambridge University Press. ISBN 0521419743.
Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory. Springer. ISBN 978-3-540-29952-3.
Fowler, David (January 1996). "The Binomial Coefficient Function". The American Mathematical Monthly (Mathematical Association of America) 103 (1): 1–17. doi:10.2307/2975209. JSTOR 2975209
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (Second ed.). Addison-Wesley. pp. 153–256. ISBN 0-201-55802-5.
Higham, Nicholas J. (1998). Handbook of writing for the mathematical sciences. SIAM. p. 25. ISBN 0898714206.
Knuth, Donald E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley. pp. 52–74. ISBN 0-201-89683-4.
Singmaster, David (1974). "Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients". J. London Math. Soc. (2) 8 (3): 555–560. doi:10.1112/jlms/s2-8.3.555.
Shilov, G. E. (1977). Linear algebra. Dover Publications. ISBN 9780486635187.
External links
Calculation of Binomial Coefficient
This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: Binomial Coefficient, Bounds for binomial coefficients, Proof that C(n,k) is an integer, Generalized binomial coefficients.
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