Editing E (mathematical constant) (section)

=== Derangements ===
{{Main|Derangement}}
Another application of {{mvar|e}}, also discovered in part by Jacob Bernoulli along with [[Pierre Remond de Montmort]], is in the problem of [[derangement]]s, also known as the ''hat check problem'':<ref>{{cite book| last1=Grinstead|first1= Charles M. |last2= Snell|first2= James Laurie |date=1997 |url=http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |title= Introduction to Probability |archive-url=https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |archive-date=2011-07-27 |type=published online under the [[GFDL]]|publisher=American Mathematical Society |page= 85 |isbn=978-0-8218-9414-9 |author2-link=J. Laurie Snell }}</ref> {{mvar|n}} guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into {{mvar|n}} boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that ''none'' of the hats gets put into the right box. This probability, denoted by <math>p_n\!</math>, is:

:<math>p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.</math>

As {{mvar|n}} tends to infinity, {{math|''p''<sub>''n''</sub>}} approaches {{math|1/''e''}}. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is {{math|''n''!/''e'',}} [[Rounding|rounded]] to the nearest integer, for every positive&nbsp;{{mvar|n}}.<ref>{{cite book | last=Knuth |first=Donald |author-link=Donald Knuth |date= 1997 | title=The Art of Computer Programming |title-link=The Art of Computer Programming | volume =I|publisher= Addison-Wesley|page= 183 |isbn=0-201-03801-3}}</ref>