Editing Derangement (section)

==Growth of number of derangements as ''n'' approaches &infin;==
From
<math display="block">!n = n! \sum_{i=0}^n \frac{(-1)^i}{i!}</math>
and
<math display="block">e^x = \sum_{i=0}^\infty {x^i \over i!}</math>
by substituting <math display="inline"> x = -1</math> one immediately obtains that
<math display="block"> \lim_{n\to\infty} {!n \over n!} = \lim_{n\to\infty} \sum_{i=0}^n \frac{(-1)^i}{i!} = e^{-1} \approx 0.367879\ldots.</math>
This is the limit of the [[probability]] that a randomly selected permutation of a large number of objects is a derangement. The probability converges to this limit extremely quickly as {{mvar|n}} increases, which is why {{math|!''n''}} is the nearest integer to {{math|''n''!/''e''.}} The above [[semi-log]] graph shows that the derangement graph lags the permutation graph by an almost constant value.

More information about this calculation and the above limit may be found in the article on the
[[Random permutation statistics#Number of permutations that are derangements|statistics of random permutations]].

=== Asymptotic expansion in terms of Bell numbers ===
An asymptotic expansion for the number of [[derangements]] in terms of [[Bell numbers]] is as follows:
<math display="block">!n = \frac{n!}{e} + \sum_{k=1}^m \left(-1\right)^{n+k-1}\frac{B_k}{n^k} + O\left(\frac{1}{n^{m+1}}\right),</math>
where <math>m</math> is any fixed positive integer, and <math>B_k</math> denotes the <math>k</math>-th [[Bell number]]. Moreover, the constant implied by the [[big O notation|big O]]-term does not exceed <math>B_{m+1}</math>.<ref>{{cite journal
 | last = Hassani  | first = Mehdi
 | year = 2020
 | title = Derangements and Alternating Sum of Permutations by Integration
 | journal = [[Journal of Integer Sequences]]
 | volume = 23 | at = Article&nbsp;20.7.8  | no-pp=yes
 | url = https://cs.uwaterloo.ca/journals/JIS/VOL23/Hassani/hassani5.html
}}</ref>