TITLE: Number of cycles in complete graph
QUESTION [7 upvotes]: How many number of cycles are there in a complete graph?
Is there any relation to Symmetric group?

REPLY [11 votes]: From $K_n$ lets pick a number $p$, $p \le n$ where $C_p$ is a cycle of length $p$.
Let $2\to 3\to 4\to 5\to 1\to\dots\to (p-1)\to 2$ be any arbitrary $C_p$. (note that the first and last vertex are same).
Now we fix the first vertex(which is also the last vertex since it is a cycle) like:
$$
2\to [3\to 4\to 5\to 1\to\dots\to (p-1)]\to 2
$$
We have $(p-1)!/2$ different cycles if we permute the section within the square brackets '$[]$' (circular permutation).
Now, from $K_n$ we can choose $p$ vertex in $\dbinom np$ ways and each of the $p$ vertex has $(p-1)!/2$ cycles.
Therefore the total number of cycles in $K_n$ is 
$$
\sum_{p=3}^{n} \binom np \frac{(p-1)!}{2}.
$$