Binomial Coefficient Calculator

About this calculator

The binomial coefficient C(n, k) = n! / (k!(n−k)!), read “n choose k,” counts the number of ways to pick an unordered subset of k items from a set of n — for example, C(5, 2) = 10 ways to choose 2 people from a group of 5. This calculator also reports the related permutation count P(n, k) = n! / (n−k)!, which counts the same selection but where order matters, so P(5, 2) = 20 since (A, B) and (B, A) count as different outcomes.

Binomial coefficients form the rows of Pascal's Triangle and the coefficients of the binomial expansion (x + y)ⁿ, and they underlie the binomial probability distribution used to calculate the odds of exactly k successes in n independent trials. Statisticians, probability students, poker and lottery-odds calculators, and combinatorics researchers rely on C(n, k) constantly — anytime a problem asks “how many ways can you choose...” without regard to order, this is the formula being applied.