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.

Hint:
A subgroup of $S_7$ of order $5$ must be cyclic (all groups of prime order are cyclic), and therefore is generated by an element of order $5$.  The only elements of order $5$ in $S_7$ are $5$-cycles (why?), but each subgroup contains $4$ such cycles.  This reduces the problem to one of a combinatorial flavor.
