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.
