9C9

9C9 is a binomial coefficient used in combinatorics to denote the number of ways to choose 9 items from a set of 9. In this case, 9C9 equals 1.

General definition and notation: nCk is defined as n! / (k!(n−k)!). Therefore, 9C9 = 9! / (9!0!) = 1. By symmetry, nCk = nC(n−k), so 9C9 = 9C0 = 1.

Applications and context: Binomial coefficients appear in the binomial theorem, where they are the coefficients in the expansion of (x + y)^n. They also arise in counting problems that involve selecting subsets of a given size from a finite set, and in probability calculations related to combinations.

Notation variants: The same concept is frequently written as C(n, k) or nCk. In software and calculators, functions may be named binom(n, k) or shown as combination notations with parentheses or braces. The term is sometimes described as “n choose k.”

Examples: From a group of 9 players, the number of ways to choose all 9 is 9C9 = 1. In the expansion of (x + y)^9, the coefficient of x^0y^9 is 9C9 = 1, illustrating how binomial coefficients index terms in the expansion.

See also: Binomial coefficient, Pascal’s triangle, combinatorics.