Binomial coefficient

History and notation

Andreas von Ettingshausen introduced the notation ( n k ) in 1826, although the numbers were known centuries earlier (see Pascal's triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingala's Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī.

Alternative notations include C(n, k), _nC_k, ⁿC_k, C^k_n, Cⁿ_k, and C_n,k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to k-permutations of n, written as P(n, k), etc.

Definition and interpretations

For natural numbers (taken to include 0) n and k, the binomial coefficient ( n k ) can be defined as the coefficient of the monomial X^k in the expansion of (1 + X)ⁿ. The same coefficient also occurs (if k ≤ n) in the binomial formula

(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 X^k, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that ( n k ) 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 ( n k ) , while the number of ways to write k = a 1 + a 2 + ⋯ + a n where every a_i is a nonnegative integer is given by ( n + k − 1 n − 1 ) . 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 ( n k ) without actually expanding a binomial power or counting k-combinations.

Recursive formula

One method uses the recursive, purely additive, formula

with initial/boundary values

The formula follows from considering the set {1,2,3,…,n} and 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 k − 1 from the remaining n − 1) 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 X^k in (1 + X)ⁿ⁻¹(1 + X). As there is zero Xⁿ⁺¹ or X⁻¹ in (1 + X)ⁿ, one might extend the definition beyond the above boundaries to include ( n k )  = 0 when either k > n or 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 formula

where the numerator of the first fraction n 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.

Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the upper limit of the product above to the smaller of k and 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 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 (in the case that k is small and 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)

which leads to a more efficient multiplicative computational routine. Using the falling factorial notation,

Generalization and connection to the binomial series

The 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 in which all positive integers are invertible:

With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the ( α k ) binomial coefficients:

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

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

Pascal's rule is the important recurrence relation

which can be used to prove by mathematical induction that ( 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:

Row number n contains the numbers ( 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

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:

Binomial coefficients as polynomials

For any nonnegative integer k, the expression ( t k ) can be simplified and defined as a polynomial divided by k!:

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 ( 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:

The derivative of ( t k ) can be calculated by logarithmic differentiation:

Binomial coefficients as a basis for the space of polynomials

Over any field of characteristic 0 (that is, any field that contains the rational numbers), each polynomial p(t) of degree at most d is uniquely expressible as a linear combination ∑ k = 0 d a k ( t k ) of binomial coefficients. The coefficient a_k is the kth difference of the sequence p(0), p(1), …, p(k). Explicitly,

Integer-valued polynomials

Each polynomial ( t k ) is integer-valued: it has an integer value at all integer inputs t . (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, (4) 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

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

and, with a little more work,

Moreover, the following may be useful:

For constant n, we have the following recurrence:

Series involving binomial coefficients

The formula

is obtained from (∗) by setting x = 1 and y = 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, (∗∗) is the statement that the power set of 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

and ∑ k = 0 n k 2 ( n k ) = ( n + n 2 ) 2 n − 2 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

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 (7) reduces to equation (3).

A similar looking formula, which applies for any integers j, k, and n satisfying 0 ≤ j ≤ k ≤ n, is

and can be found by examination of the coefficient of x n + 1 in the expansion of

using

When j = k, equation (8) gives the Hockey-stick identity

From expansion (7) using n = 2m, k = m, and (1), one finds

Let F(n) denote the n-th Fibonacci number. We obtain a formula about the diagonals of Pascal's triangle,

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)). A combinatorial proof is given below.

Another identity that follows from (8) with j=k-1 is

Partial sums

Although there is no closed formula for partial sums

of binomial coefficients, one can again use (3) and induction to show that for k = 0, ..., n − 1,

with special case

for n > 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,

Differentiating (2) k times and setting x = −1 yields this for P ( x ) = x ( x − 1 ) ⋯ ( 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,

where a n is the coefficient of degree n in P(x).

More generally for (10),

where m and d are complex numbers. This follows immediately applying (10) 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 dⁿa_n.

The series k − 1 k ∑ j = 0 ∞ 1 ( j + x k ) = 1 ( x − 1 k − 1 ) is convergent for k ≥ 2. This formula is used in the analysis of the German tank problem. It follows from k − 1 k ∑ j = 0 M 1 ( j + x k ) = 1 ( x − 1 k − 1 ) − 1 ( M + x k − 1 ) which is proved by induction on M.

Using (9) one can derive ∑ i = 0 n i ( n i ) 2 = n 2 ( 2 n n ) and ∑ i = 0 n i 2 ( n i ) 2 = n 2 ( 2 n − 2 n − 1 ) .

Series multisection gives the following identity for the sum of binomial coefficients taken with a step s and offset t ( 0 ⩽ t < s ) as a closed-form sum of s terms:

Identities with combinatorial proofs

Many identities involving binomial coefficients can be proved by combinatorial means. For example, the following identity for nonnegative integers n ≥ q (which reduces to (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 parameter, because there are ( 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

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 (9) also has a combinatorial proof. The identity reads

Suppose you have 2 n empty squares arranged in a row and you want to mark (select) n of them. There are ( 2 n 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 (5) to get the result.

The identity (9),

has the following combinatorial proof. The number ( n − k k ) denotes the number of paths in a two-dimensional lattice from ( 0 , 0 ) to ( k , n − k ) using steps ( 0 , 1 ) and ( 1 , 1 ) . This is easy to see: there are ( n − k ) steps in total and one may choose the k ( 1 , 1 ) steps. Now, replace each ( 1 , 1 ) step by a ( 0 , 2 ) step; note that there are exactly k . Then one arrives at point ( 0 , n ) using steps ( 0 , 1 ) and ( 0 , 2 ) . Doing this for all k between 0 and ⌊ n 2 ⌋ gives all paths from ( 0 , 0 ) to ( 0 , n ) using steps ( 0 , 1 ) and ( 0 , 2 ) . Clearly, there are exactly F ( n + 1 ) such paths.

Sum of coefficients row

The number of k-combinations for all k, ∑ 0 ≤ k ≤ n ( n k ) = 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 m , n ∈ N

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.

Ordinary generating functions

For a fixed n, the ordinary generating function of the sequence ( n 0 ) , ( n 1 ) , ( n 2 ) , … is:

For a fixed k, the ordinary generating function of the sequence ( 0 k ) , ( 1 k ) , ( 2 k ) , … is:

The bivariate generating function of the binomial coefficients is:

Another bivariate generating function of the binomial coefficients, which is symmetric, is:

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 ( m + n m ) equals p^c, where c is the number of carries when m and n are added in base p. Equivalently, the exponent of a prime p in ( n k ) equals 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 ( n k ) is divisible by n/gcd(n,k). In particular therefore it follows that p divides ( p r s ) for all positive integers r and s such that s < p^r. However this is not true of higher powers of p: for example 9 does not divide ( 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 ( n k ) with n < N such that d divides ( n k ) . Then

Since the number of binomial coefficients ( 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

are divisible by n.

Proof: When p is prime, p divides

because ( 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 k = n/p. Then 0 < p < n and

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 ( n k ) hold for all values of n and k such that for 1 ≤ k ≤ n:

Stirling's approximation yields the bounds:

and the approximation

For both n and k much larger than 1, Stirling's approximation also yields the following asymptotic approximation:

where H ( ϵ ) = − ϵ log 2 ⁡ ( ϵ ) − ( 1 − ϵ ) log 2 ⁡ ( 1 − ϵ ) is the binary entropy of ϵ . More precisely, for all integers n ≥ k ≥ 1 with ϵ ≐ k / n ≤ 1 / 2 , we can estimate the sum of the first k + 1 binomial coefficients as follows:

When n is large and k is much smaller than n , one can also write

and therefore

If more precision is desired, one can approximate ln ⁡ ( n ( n − 1 ) … ( n − k + 1 ) ) with an integral, obtaining

For n = 20 and k = 10 , log ⁡ ( n k ) ≈ 12.127 , and these approximations yield 12.312 and 12.133 respectively.

The infinite product formula (cf. Gamma function, alternative definition)

yields the asymptotic formulas

as k → ∞ .

This asymptotic behaviour is contained in the approximation

as well. (Here H k is the k-th harmonic number and γ is the Euler–Mascheroni constant.)

Further, the asymptotic formula

hold true, whenever k → ∞ and j / k → x for some complex number x .

A simple and rough upper bound for the sum of binomial coefficients can be obtained using the binomial theorem:

If n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient ( n k ) . For example, if | n / 2 − k | = o ( n 2 / 3 ) then

where d = n − 2k.

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 (x+y)ⁿ, the multinomial coefficients represent 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 k_i 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:

and symmetry:

where ( σ 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 n = ½

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 1 + x into a power series using the Newton binomial series :

Identity for the product of binomial coefficients

One can express the product of 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 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 reciprocal is given by

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 (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; the number of ways to "multichoose" (i.e., choose with replacement) k items from an n element set is denoted ( ( n k ) ) .

To avoid ambiguity and confusion with n’s main denotation in this article,
let f = n = r + (k – 1) and r = f – (k – 1).

Multiset coefficients may be expressed in terms of binomial coefficients by the rule

One possible alternative characterization of this identity is as follows: We may define the falling factorial as

and the corresponding rising factorial as

so, for example,

Then the binomial coefficients may be written as

while the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:

Generalization to negative integers

For any n,

In particular, binomial coefficients evaluated at negative integers are given by signed multiset coefficients. In the special case n = − 1 , this reduces to ( − 1 ) k = ( − 1 k ) = ( ( − k k ) ) .

For example, if n = -4 and k = 7, then r = 4 and f = 10:

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 Γ :

moreover,

The resulting function has been little-studied, apparently first being graphed in (Fowler 1996). Notably, many binomial identities fail: ( n m ) = ( n n − m ) but ( − n m ) = ( − 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:

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 A is some set with cardinality α . 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 α , ( α β ) 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 ( α α ) = 2 α for any infinite cardinal α .

Binomial coefficient in programming languages

The notation ( n 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 both the APL programming language and the (related) J programming language use the exclamation mark: k ! n .

Naive implementations of the factorial formula, such as the following snippet in Python:

are very slow and are useless for calculating factorials of very high numbers (in languages such as C or Java they suffer from overflow errors because of this reason). A direct implementation of the multiplicative formula works well:

(In Python, range(k) produces a list from 0 to k–1.)

Pascal's rule provides a recursive definition which can also be implemented in Python, although it is less efficient:

The example mentioned above can be also written in functional style. The following Scheme example uses the recursive definition

Rational arithmetic can be easily avoided using integer division

The following implementation uses all these ideas

When computing ( n k + 1 ) = [ ( n − k ) ( n k ) ] ÷ ( k + 1 ) in a language with fixed-length integers, the multiplication by ( n − k ) may overflow even when the result would fit. The overflow can be avoided by dividing first and fixing the result using the remainder:

Implementation in the C language:

Another way to compute the binomial coefficient when using large numbers is to recognize that

where ln   Γ ( 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, gammaln in MATLAB and Python's SciPy module, or lgamma in R and Julia. Roundoff error may cause the returned value to not be an integer.