nCnk

nCnk, commonly written as nCk or C(n, k), denotes the binomial coefficient, the number of ways to choose k elements from a set of n distinct elements. It is defined for nonnegative integers n and k with 0 ≤ k ≤ n by the formula nCk = n! / (k!(n−k)!). When k < 0 or k > n, nCk is defined to be 0. The binomial coefficient can also be expressed using the gamma function as Γ(n+1) / [Γ(k+1) Γ(n−k+1)], which allows extension to non-integer n in a standard way.

Key properties include the symmetry nCk = nC(n−k), reflecting that choosing k elements is equivalent to omitting n−k elements. A common recurrence is nCk = (n−1)Ck + (n−1)C(k−1), with base cases nC0 = nCn = 1 and nC1 = n. These relations underpin Pascal’s triangle, where each entry is the sum of the two above.

Combinatorial interpretation is central: nCk counts the number of k-element subsets of an n-element set. It also appears in probability and statistics, such as in binomial distributions, and in algebra via the binomial theorem: (x + y)^n = sum_{k=0}^n nCk x^k y^{n−k}.

Special values include nC0 = 1 and nCn = 1, and nC1 = n. For integers n ≥ 0, nCk is zero outside the range 0 ≤ k ≤ n. Generalizations include q-binomial coefficients and other extensions used in combinatorics and representation theory. Example calculations: 5C2 = 10; 6C3 = 20.