I am unfamiliar with number theory but am trying to calculate the following for a coding challenge:
$$\frac{(N-M-1)!}{N!(M-1)!}\pmod{Q}$$
where $Q$ is prime. I know that I can calculate the "modular" factorial fairly easily:
$$(N-M-1)!\bmod{Q} \equiv (((N-M-1)\times (N-M-2))\bmod{Q} \times \dots))\bmod{Q}$$
However, I'm not sure how I calculate the division part, I know that we can rewrite the fraction:
$$\frac{(N+M-1)!}{N!(M-1)!}\equiv (N+M-1)!\cdot (N!(M-1)!)^{-1}\pmod{Q}$$
However, can I compute the "modular" factorial of $N$ and $(M-1)$ in the same way and then compute the modular inverse of their product, or is there something else I need to do?