# Binomial coefficient

In mathematics, the **binomial coefficients** are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers *n* ≥ *k* ≥ 0 and is written It is the coefficient of the *x*^{k} term in the polynomial expansion of the binomial power (1 + *x*)^{n}, and is given by the formula

For example, the fourth power of 1 + *x* is

and the binomial coefficient is the coefficient of the *x*^{2} term.

Arranging the numbers in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation

The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol is usually read as "*n* choose *k*" because there are ways to choose an (unordered) subset of *k* elements from a fixed set of *n* elements. For example, there are ways to choose 2 elements from namely and

The binomial coefficients can be generalized to for any complex number z and integer *k* ≥ 0, and many of their properties continue to hold in this more general form.