Montmort matching problem

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derangement problem

Take two sets, and , and a bijection, , between them. (E.g., take married couples and let be the set of husbands and the set of wives.) Now, take a random paring (a bijection again). What is the chance that this random pairing gives at least one "correct match" (i.e. coincides with in at least one element). Asymptotically, this probability is . This follows immediately from the formula given in Classical combinatorial problems for the number of permutations such that for all .

This problem was considered first by P.R. de Montmort (around 1700) in connection with a card game known as the "jeu du treize" , "jeu de rencontre" or simply "rencontre" .


[a1] E. Knobloch, "Euler and the history of a problem in probability theory" Ganita–Bharati , 6 (1984) pp. 1–12
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Montmort matching problem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article