Consider a sequence of n independent trials, each of which can result in an outcome in any of k categories. Let pj be the probability that each trial results in an outcome in category j, j = 1, 2, … , k, so

p₁ + p₂ + … + pk = 100%.

The number of outcomes of each type has a multinomial distribution. In particular, the probability that the n trials result in n₁ outcomes of type 1, n₂ outcomes of type 2, … , and nk outcomes of type k is

n!/(n₁! × n₂! × … × nk!) × p₁n₁ × p₂n₂ × … × pknk,

if n₁, … , nk are nonnegative integers that sum to n; the chance is zero otherwise.