PIERRE REMOND DE MONTMORT’S MATCHING PROBLEM 1708

In its simplest form the problem of coincidences (matches, recontres) may be formulated as follows: Let there be n objects numbered from 1 to n, and let them be ordered at random, assuming that the n! permutations are equally probable. A coincidence occurs if object number i is found at the ith place. The problem is to find the number of permutations with at least one coincidence or, equivalently, the probability of at least one coincidence.
When the sample size is 5, what’s the probability of observing at least one match? For n = 5, the argument is as follows: The number of permutations of 5 cards is 5! = 120. Among these there are 24 in which 1 is in first place, 18 in which 2 is in second place without 1 being first, 14 in which 3 is in third place without 1 being first or 2 being second, 11 in which 4 is in fourth place without 1 being first, 2 being second or 3 being third, and, finally, 9 in which 5 is in fifth place, the other four being out of their places. The probability of at least one coincidence is therefore:[A. Hald]