Derangement
A derangement is a permutation with no fixed points. That is, a derangement of a set leaves no element in its original place. For example, the derangements of are
and
but not
because 2 is a fixed point.
The number of derangements of a set of objects is sometimes denoted
and is given by the formula
![$!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$](http://latex.artofproblemsolving.com/8/2/8/82863216c3e5a66357119aa2b0fd720d478c2ea5.png)
Thus, the number derangements of a 3-element set is , which we know to be correct.
Introductory
See also
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