How many $5$-element subgroups does $S_7$ have? 
How many $5$-element subgroups are there in $S_7$, the group of permutations on $7$ elements?

Let $H$ be a $5$-element subgroup of $S_7$. We have $\mbox{ord}(H) = 5$ and $5\mid 7!$. 
But I don't have any idea how can I find 5-element subgroups.

Hints (completing and/or adding to some other comments or answers):


*

*Each group of order $\;5\;$ in $\;S_7\;$ is generated by a $\,5-$cycle (this is false in $\,S_n\;,\;\;n\ge10\;$ . Can you see why?)

*For each given subset of five elements in $\;\{1,2,3,4,5,6,7\}\;$ we get a generator of a subgroup of order $\;5\;$ in $\;S_7\;$ and all of them yield different subgroup of order $\;5\;$ in $\;S_7\;$ .

*(kind of) Another approach: How many $\;5\;$ cycles are there in $\,S_7\;$ ? How many of them generate the same cyclic subgroup? For example
$$\langle (12345)\rangle =\langle(13524)\rangle=\langle(14253)\rangle =\ldots$$
