The Subfactorial is a variant of a factorial, denoted as ${\displaystyle !n}$. The subfactorial of n is the amount of permutations in which no element appears in its original position, also known as the number of derangements of a number.

The formula for calculating ${\displaystyle !n}$ is:

${\displaystyle !n=n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}}$
where n is greater than or equal to 0.

Another way to find ${\displaystyle !n}$ is:

${\displaystyle !n=\left\lfloor {\frac {n!}{e}}\right\rceil }$

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