Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probability pi , i = 1, . . . , r, r i=1 pi = 1. If n of these experiments are performed, and if the outcome of any one of the n does not affect the outcome of the other n−1 experiments, then show that the probability that the first outcome appears x1 times, the second x2 times, and the rth xr times is

This is known as the multinomial distribution.

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n! P1 P2 P when x1+x2 + + xr = n X1!x2!...xp!