I have reached a brick wall with a specific issue regarding the Travelling Salesman Problem (TSP). I need to calculate the number of possible routes between 10 locations, so I can find out the time it takes for a brute force algorithm to find the shortest one of all. Through some calculations I have reached the conclusion that there is a total of 1, 814, 400 possible routes. However, I have read that there are actually 181, 440 possible routes (ScienceDirect, https://www.sciencedirect.com/topics/earth-and-planetary-sciences/traveling-salesman-problem). I believe that 1, 814, 400 is the correct answer (As according to Marcus du Sautoy, https://www.bbc.co.uk/programmes/p030s6b3) but I am not sure, I have searched and found contradicting information.
This is my first question, I have tried to provide all the necessary information, correct formatting etc. but if I need to adjust anything then please let me know. Thanks
There are $10!$ routes, but the starting point of the cycle is irrelevant, so you have to divide by 10 for the cyclic symmetry, and then it overcounts by two because there are "anticlockwise" and "clockwise" versions of the same route, so the number of unique roots is $$ {10!\over 20}=181,440, $$ so neither of those numbers is correct.
I clicked on the link you provided to support the 181,400 value, and note that the formula there agrees with me, and it doesn't give the value 181,400, so I assume that was your calculation error.