Subfactorial

The Subfactorial is a variant of a factorial, denoted as $!n$. The subfactorial of n is the amount of permutations in which no element appears in its original position.

The formula for calculating $!n$ is:

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