What is "binomial coefficient"

n. (context combinatorics English) A coefficient of any of the terms in the expansion of the binomial (''x''+''y'')^''n''.

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 $\tbinom nk$. It is the coefficient of the x term in the polynomial expansion of the binomial power (1 + x). Under suitable circumstances the value of the coefficient is given by the expression $\tfrac{n!}{k!\,(n-k)!}$. 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 areas of mathematics other than algebra, especially in combinatorics. $\tbinom nk$ is often read aloud as "n choose k", because there are $\tbinom nk$ ways to choose k elements, disregarding their order, from a set of n elements. 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 ; such expressions are still called binomial coefficients.