Thinking Mathematically (6th Edition)

The original statement does not make sense. The number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$. If a complete graph has 5 vertices, then the number of Hamilton circuits is $(5-1)! = 4! = 24$. If a complete graph has 6 vertices, then the number of Hamilton circuits is $(6-1)! = 5! = 120$. There is no whole number $n$ such that $(n-1)! = 25$. Therefore, there is no complete graph which has 25 Hamilton circuits.