binomialnk

Binomialnk is a mathematical term used to denote the binomial coefficient, the number of ways to choose k elements from a set of n distinct elements without regard to order. It is commonly written as C(n, k) or binomial(n, k). The standard definition applies to integers n and k with 0 ≤ k ≤ n, and is given by binomialnk = n! / (k!(n−k)!). If k < 0 or k > n, binomialnk is defined to be 0.

Key properties include symmetry, binomialnk = binomial(n, n−k), which reflects the equivalence of choosing k items or omitting n−k items. Another fundamental relation is Pascal’s rule: binomial(n, k) = binomial(n−1, k−1) + binomial(n−1, k). This recursive property underpins constructions such as Pascal’s triangle.

There are several practical methods to compute binomialnk. The factorial formula is direct but can be inefficient for large n due to large intermediate values. A multiplicative formula, binomial(n, k) = product from i=1 to k of (n−k+i)/i, reduces intermediate size and is commonly used in programming and numerical libraries. Dynamic programming can also be employed for repeated calculations across ranges of n and k.

Applications of binomialnk span combinatorics, probability, and statistics. They appear as the coefficients in the expansion of (x + y)^n, in binomial distributions, and in generating functions. Example: binomial(5, 2) = 10.

See also: binomial coefficient, binomial theorem, Pascal’s triangle, multinomial coefficient.