Number of Hamilton Cycles in Complete Graph

Theorem
For all $$n \ge 3$$, the number of distinct Hamilton cycles in the complete graph $$K_n$$ is $$\frac {\left({n-1}\right)!} 2$$.

Proof
In a complete graph, every vertex is adjacent to every other vertex.

Therefore, if we were to take all the vertices in a complete graph in any order, there will be a path through those vertices in that order.

Joining either end of that path gives us a Hamilton cycle.

From Cardinality of Set of Bijections, there are $$n!$$ different ways of picking the vertices of $$G$$ in some order.

Hence there are $$n!$$ ways of building such a Hamilton cycle.

Not all these are different, though.

On any such cycle, there are:
 * $$n$$ different places you can start;
 * $$2$$ different directions you can travel.

So any one of these $$n!$$ cycles is in a set of $$2n$$ cycles which all contain the same set of edges.

So there are $$\frac {n!} {2n} = \frac {\left({n-1}\right)!} 2$$ distinct Hamilton cycles.