I've been given the following problem as homework:
Q: Compute the number of subgraphs of $K_{15}$ isomorphic to $C_{15}$.
$K_{15}$ means complete graph with 15 vertices. $C_{15}$ means cyclic graph, where the whole graph is a cycle, with 15 vertices. For example, $C_3$ is a triangle, $C_4$ is a square, and $C_5$ is a pentagon.
My efforts: In order to try to figure out a general formula for $C_n$, I tried doing this problem with $C_5$. After a huge amount of trial and error, it looks like the formula is something like $\binom{15}{n}\binom{n-1}{2}(n-3)!$
However, I can't seem to come up with a good reason for this formula or verify whether it's correct. I'm doubting it is correct.
This is homework, so I'm NOT looking for solutions. Could you give some tips for figuring this out?