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A derangement is a permutation with no fixed points. A derangement can also be thought of as a permutation in which none of the objects are in their original space. For example, the derangements of $(1,2,3)$ are $(2, 3, 1)$ and $(3, 1, 2)$. The number of derangements of a set of x objects is denoted !x, and is given by the formula:

$\displaystyle !x = x! \sum_{k=1}^{n} \frac{-1^k}{k!}$

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