Count of All Permutations on n Objects

Theorem
Let $S$ be a set of $n$ objects.

Let $N$ be the number of permutations of $k$ objects from $S$, where $1 \le k \le N$.

Then:
 * $\ds N = n! \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {k!}$

Proof
The number of permutations on $k$ objects, from $n$ is denoted ${}^n P_k$.

From Number of Permutations:


 * ${}^n P_k = \dfrac {n!} {\paren {n - k}!}$

Hence: