In this problem, we employ the multinomial distribution to solve an extension of the birthday problem. Assuming that each of n individuals is, independently of others, equally likely to have their birthday be any of the 365 days of the year, we want to derive an expression for the probability that the collection of n individuals will contain at least 3 having the same birthday.

(a) For a given partition of the 365 days of the year into a first set of size i, a second set of size n − 2i and a third of size 365 − n + i, find the probability that every day in the first set is the birthday of exactly 2 of the n individuals, every day in the second set is the birthday of exactly 1 of the n individuals, and every day in the third set is the birthday of none of the n individuals.

(b) For a given value i, determine the number of different partitions of the 365 days of the year into a first set of size i, a second set of size n − 2i and a third set of size 365− n+i.

(c) Give an expression for the probability that a collection of n individuals does not contain at least 3 having the same birthday.

Remark. A computation gives that

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44 1- i=0 365! 88! 1 88 0.504. i!(88 2i)!(365-88+i)! 2 365 -