The subfactorial or left factorial, written \(!n\), is the number of ways that n objects can be arranged where no object appears in its natural position (known as "derangements.") There are many formulas for \(!n\): \begin{eqnarray*} !n &=& n! \displaystyle\sum^{n}_{i = 0} \frac{(-1)^i}{i!}\\ &=& \displaystyle\sum^{n}_{i = 0} i! (-1)^{n - i} \binom{n}{i}\\ &=& \displaystyle\frac{\Gamma(n + 1, -1)}{e}\\ &=& \left[\frac{n!}{e}ight] ext{ (only for $n > 0$)} \end{eqnarray*} In the last formula, [n] means the nearest integer to n. (It is a direct consequence of the first formula — the summation converges to \(1/e\).)