## Thinking Mathematically (6th Edition)

The original statement does not make sense. The number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$. For example, if a complete graph has 6 vertices, then the number of Hamilton circuits is $(6-1)! = 5! = 120$. We have not yet been able to find a simple method to determine the optimal Hamilton circuit for a weighted graph. One method we can use is the Brute Force Method, which involves listing all the possible Hamilton circuits and calculating the total weight of each Hamilton circuit. For a graph with a small number of vertices, the Brute Force Method is a reasonable method. However, if a graph has a large number of vertices, the number of Hamilton circuits can be extremely large. Even for a super-computer, it would be impractical to list all the possible Hamilton circuits and calculate the total weight of each Hamilton circuit. If a complete graph has 20 vertices, then the number of Hamilton circuits is $(20-1)! = 19!$ which is an enormous number. Even for a super-computer, it would take much longer than one evening to list all the possible Hamilton circuits and calculate the total weight of each Hamilton circuit.