Number of k-Cycles in Symmetric Group/Proof 2

Proof
Suppose $n \ge k$, and consider the number of $k$-cycles in $S_n$.

A $k$-cycle can be represented by a selection of $k$ elements from $n$ without any repeats.

From Number of Permutations, the number of permutations of $k$ elements from $n$ possible elements is $\dfrac {n!} {\paren {n - k}!}$.

However, each such string is merely a representation of an $k$-cycle; the $k$-cycle itself does not depend on the starting elements in the string.

Since there are $k$ possible starting elements, we must divide this number by $k$.

Hence, the number of $m$-cycles is
 * $\dfrac {n!} {k \paren {n - k}!}$