Answer
This graph has 39,916,800 Hamilton circuits.
Work Step by Step
This complete graph has 12 vertices. We know that the number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$.
The number of Hamilton circuits in this graph is $(12-1)!$ which is $11!$. Since $11! = 39,916,800$, this graph has 39,916,800 Hamilton circuits.