View source for Montmort matching problem

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''derangement problem''

Take two sets, $A$ and $B$, and a [[bijection]], $\phi$, between them. (E.g., take $n$ married couples and let $A$ be the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at least one  "correct match"  (i.e. coincides with $\phi$ in at least one element). Asymptotically, this [[probability]] is $1-e^{-1}$. This follows immediately from the formula given in [[Classical combinatorial problems|Classical combinatorial problems]] for the number of ''[[derangement]]s'': permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$.

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".

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Knobloch,   "Euler and the history of a problem in probability theory"  ''Ganita–Bharati'' , '''6'''  (1984)  pp. 1–12</TD></TR></table>