![Let ๐บ be an undirected complete graph on ๐ vertices, where ๐ > 2. Then, the number of different Hamiltonian cycles in ๐บ is equal to](https://www.gkseries.com/blog/wp-content/uploads/2023/08/Let-G-be-an-undirected-complete-graph-on-๐-vertices-where-๐-2.-Then-the-number-of-different-Hamiltonian-cycles-in-๐บ-is-equal-to.jpg)
Q. Let ๐บ be an undirected complete graph on ๐ vertices, where ๐ > 2. Then, the number of different Hamiltonian cycles in ๐บ is equal to
![](https://www.gkseries.com/blog/wp-content/uploads/2023/08/Screenshot-484.png)
Ans: Option D
Solution:
A simple circuit in a graph G that passes through every vertex exactly once is called a Hamiltonian circuit. In Hamiltonian cycle is a Hamiltonian circuit in which initial and final vertex are same. In an undirected complete graph on n vertices, there are n permutations are possible to visit every node. But from these permutations, there are: n different places (i.e., nodes) you can start; two (clockwise or anticlockwise) 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 =
![](https://www.gkseries.com/blog/wp-content/uploads/2023/08/Screenshot-485.png)
distinct Hamilton cycles.