This question already has an answer here:
I have seen multiple times that physicists used a formula to find the number of vacua in string theory, What formula was used to calculate the number of vacua in string theory? How did they arrive at 10^500? This is somewhat different from the question of why string theory has such a large landscape; I am asking for the mathematical details of the situation not just the logic. I want to know what math led to this, not "because there are a lot of manifolds"