A note on the “probleme des rencontres.”

Abstract

1. This celebrated problem is treated in nearly all the textbooks on probability; for example in Bertrand's Calcul des Probabilités, 1889, pp. 15–17, in Poincare's of the same title, 1896, pp. 36–38, and in most of the recent textbooks. The problem may be stated in abstract terms as follows: Among the n! permutations (α1α2α3… αn) of the natural order (123…n), how many have no αi equal to j? The problem has been clothed in many picturesque (and highly unlikely) “representations”; for example, by imagining n letters placed at random in n addressed envelopes, and inquiring what is the chance that no letter is in its correct envelope; or by imagining n gentlemen returning at random to their n houses; and so on, ad risum. Various derivations have also been given of the probability in question, namely the first n + 1 terms of the expansion of e-1, to which function the probability converges with rapidity as n increases.