Count of All Permutations on n Objects

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

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

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

Proof
The number of permutations on $k$ objects, from $n$ is denoted ${}^k P_{10}$.

From Number of Permutations:


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

Hence: