If n objects are distributed at random among n persons, the probability that at least one of them will not get anything is

The first object can be given to any of the n persons, but the second, third and other objects; too, can go to any of the n persons. Therefore the total number of ways of distributing the n objects randomly among n persons isnn . There are nPn=n! ways in which each person gets exactly one object, so the probability of this happening is n!nn=(n−1)!nn−1 .Hence the probability that at least one person does not get any object is        =1−(n−1)!nn−1