Traveling Salesman Number of Possible Routes

Question

My question is:

If there are 12 cities to visit, how many possible routes are?

Are there (11*10*9*8*7*6*5*4*3*2*1)/2 = 19,958,400 routes?

Answer 1

Basically, that is correct, but...

Why do you start with 11? The salesman has 12 cities to start with, so the product would have a 12* in the beginning.

Except if the question says 'starting in a given city', but then it's effectively a 11 city problem, not 12.

Answer 2

You are correct.

The traveling salesman problem with $n$ cities has

$\frac{(n-1)!}{2}$

routes.

It is $(n-1)!$ instead of $n!$ because it does not matter in which city you start.