Alternative notations include C(n, k), nCk, nCk, Ckn, Cnk, and Cn,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.
(valid for any elements x, y of a commutative ring), which explains the name "binomial coefficient".
A more efficient method to compute individual binomial coefficients is given by the formula
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.
Finally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar factorial function:
which leads to a more efficient multiplicative computational routine. Using the falling factorial notation,
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:
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 power 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.
Binomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:
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.
Its coefficients are expressible in terms of Stirling numbers of the first kind:
The factorial formula facilitates relating nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, then
follow from the binomial theorem after differentiating with respect to x (twice for the latter) and then substituting x = y = 1.
The Chu–Vandermonde identity, which holds for any complex-values m and n and any non-negative integer k, is
Another form of the Chu–Vandermonde identity, which applies for any integers j, k, and n satisfying 0 ≤ j ≤ k ≤ n, is
For small s, these series have particularly nice forms; for example,
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.
If one denotes by F(i) the sequence of Fibonacci numbers, indexed so that F(0) = F(1) = 1, then the identity
A symmetric bivariate generating function of the binomial coefficients is
A symmetric exponential bivariate generating function of the binomial coefficients is:
Binomial coefficients have divisibility properties related to least common multiples of consecutive integers. For example:
Another fact: An integer n ≥ 2 is prime if and only if all the intermediate binomial coefficients
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.
A simple and rough upper bound for the sum of binomial coefficients can be obtained using the binomial theorem:
The infinite product formula for the Gamma function also gives an expression for binomial coefficients
Binomial coefficients can be generalized to multinomial coefficients defined to be the number:
While the binomial coefficients represent the coefficients of (x+y)n, the multinomial coefficients represent the coefficients of the polynomial
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:
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 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.
The product of all binomial coefficients in the nth row of the Pascal triangle is given by the formula:
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
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
The definition of the binomial coefficient can be generalized to infinite cardinals by defining:
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:
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
Another way to compute the binomial coefficient when using large numbers is to recognize that