It is easy to check that we can embed ( injective homomorphism ) $C_3$ in $S_3$, where $C_3$ is a cyclic group of order 3. Just map a order 3 element to order 3 element of $S_3$.
Same idea can be extended to embed $C_n$ into $S_n$, take the generator of $C_n$ and map it into an element of order $n$ in $S_n$.
Question : How many injective homomorphisms are there from $C_n$ to $S_n$?
To me answer seems equal to number of elements of order $n$ in $S_n$