# Binomial Coefficient

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 . It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n. Under suitable circumstances the value of the coefficient is given by the expression . 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 . Therefore is often read as "n choose k". The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with kn; such expressions are then still called binomial coefficients.

The notation was introduced by Andreas von Ettingshausen in 1826, 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 Indian mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book Lilavati.

Alternative notations include C(n, k), nCk, nCk, Ckn, Cnk, in all of which the C stands for combinations or choices.

### Other articles related to "binomial coefficients, binomial coefficient, coefficients, binomial":

Freshman's Dream - Prime Characteristic
... This can be seen by examining the prime factors of the binomial coefficients the nth binomial coefficient is The numerator is p factorial, which is divisible by p ... Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring ... We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation ...
List Of Factorial And Binomial Topics
... This is a list of factorial and binomial topics in mathematics, by Wikipedia page ... See also binomial (disambiguation) ... Abel's binomial theorem Alternating factorial Antichain Beta function Binomial coefficient Binomial distribution Binomial proportion confidence interval Binomial-QMF (Daubechies wavelet filters) Binomial ...
Binomial Coefficient in Programming Languages
... a standard subroutine for computing the binomial coefficient, but for example the J programming language uses the exclamation mark k ! n ... 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 ...
Integer Partition - Partition Function - Generating Function - Gaussian Binomial Coefficient
... The Gaussian binomial coefficient is related to integer partitions ... The Gaussian binomial coefficient is defined as The number of integer partitions that would fit into a k by l rectangle (when expressed as a Ferrers or Young diagram ... The Gaussian binomial coefficient is related to the generating function of p(n, k, l) by the following equality ...
Field With One Element - Computations
... Subsets are subspaces The binomial coefficient gives the number of m-element subsets of an n-element set, and the q-binomial coefficient gives the number of m-dimensional ... The expansion of the q-binomial coefficient into a sum of powers of q corresponds to the Schubert cell decomposition of the Grassmannian ...