## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.3 Hamilton Paths and Hamilton Circuits - Exercise Set 14.3 - Page 923: 42

#### Answer

There are 120 different routes which start at home and end at home.

#### Work Step by Step

We know that the number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$. This complete graph has 6 vertices. The number of Hamilton circuits in this graph is $(6-1)! = 5! = 120$. This graph has 120 Hamilton circuits. Therefore, there are 120 different routes which start at home and end at home.

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