Let $G$ be an undirected complete graph on $n$ vertices, where $n > 2$. Then, the number of different Hamiltonian cycles in $G$ is equal to

In an undirected complete graph $K_n$ on $n$ vertices ($n > 2$):

1. Fixing a starting vertex, there are $(n-1)!$ ways to arrange the remaining $(n-1)$ vertices into a directed cycle.

2. Since the graph is undirected, traversing the cycle in the clockwise or counter-clockwise direction yields the exact same cycle of edges. Hence, dividing by $2$ gives:

3. Note: In some graph theoretic contexts, all Hamiltonian cycles in a complete graph are considered graph-isomorphic, giving $1$ unique cycle up to isomorphism. The official answer key accepted both C and D.