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 k ≤ n; 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), _nC_k, nC_k, Ck_n, Cn_k, in all of which the C stands for combinations or choices.