NCN

nCn, read as "n choose n," is the binomial coefficient with parameters n and n. In combinatorics, it represents the number of ways to select n elements from a set of n elements. For non-negative integers n, the value is computed as nCn = n! / (n! (n − n)!) = n! / (n! 0!) = 1, since 0! = 1. Consequently, nCn = 1 for every n ≥ 0, and 0C0 is also defined as 1.

In Pascal's triangle, each row begins and ends with 1, reflecting that C(n,0) = C(n,n) = 1. In the binomial theorem, (x + y)^n = sum_{k=0}^n C(n,k) x^{n−k} y^k; the term with k = n is C(n,n) y^n, which equals y^n, consistent with the identity above.

Note on generalization: The standard definition of binomial coefficients requires integer n and k with 0 ≤ k ≤ n. For non-integer n, binomial coefficients are defined via the generalized formula C(n,k) = n(n−1)...(n−k+1)/k!, which remains meaningful for integer k; setting k = n still yields 1 when n is a non-negative integer. In other contexts, C(n,n) is not defined if n is not a non-negative integer.