Why isn't the search space size of TSP $(n-1)!$ instead of $n!$?
The search space has size $(n-1)!$. You can see this in many ways, for example:


Each tour corresponds to $n$ different permutations, depending on the starting point.
If we start describing the tour at some point $P$ fixed ahead of time, then the tour is given (injectively) by an arbtirary permutation of the remaining $n-1$ points.


The reason that sometimes $n!$ is used is that $n!$ is shorter, and the difference between $n!$ and $(n-1)!$ is not always important.