Derangements and the hat-check problem

Solved in 1713 by Pierre Rémond de Montmort and Nicholas Bernoulli.

Montmort considered the problem in his 1708 essay on games of chance and solved it in 1713, as did Nicholas Bernoulli at about the same time. Inclusion and exclusion over the set of guests who do get their own hats gives $p_n={\sum }_{i=0}^n(-1{)}^i/i!$. This converges very quickly to $1/e\approx 0.3679$, so the answer barely depends on $n$ once $n$ is moderately large. The number of derangements, $D_n=n!p_n$, is the integer closest to $n!/e$ for every $n\ge 1$.

References

• Derangement (Wikipedia)