The Poisson Variation of Montmort's Matching Problem

Known as le probleme des rencontres, Montmort's problem has become a classic. Treated in almost every modern text on combinatorics, it has been recast in a variety of amusing guises including the random switching of n hats by a possibly tipsy cloakroom attendant, the random placement of n letters in n envelopes by a secretary apparently having a bad day, and even in terms of a wager on the ordering in which the horses finish in a given race (for instance, see [3], [4], and [8]). As is well known (e.g., see [2, p. 107]), the principle of inclusion-exclusion may be used to show that the probability M, k of k matches occurring with an n-card deck is