Derangement

A derangement is a permutation with no fixed points. That is, a derangement leaves nothing in its original place. For example, the derangements of $(1,2,3)$ are $(2, 3, 1)$ and $(3, 1, 2)$ .

The number of derangements of a set of $n$ objects is sometimes denoted $!n$ and is given by the formula:

$\displaystyle !n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}$

Thus, the number derangements of a 3-element set is $3! \cdot \sum_{k = 0}^3 \frac{(-1)^k}{k!} = 6\cdot(\frac{1}{1} - \frac{1}{1} + \frac{1}{2} - \frac{1}{6}) = 2$ , which we know to be correct.

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Derangement