I want to prove the following equation but don't know how to get rid of these factorials.
Are you sure you have to prove it? Try picking a few random numbers: $\displaystyle
\begin{cases} n = 3 & m = 2 \\ p = 1 & q = 2 \end{cases}$
$\displaystyle \frac{n!}{(m-1)!(n-m+1)!}p^{m-1}q^{n-m+1} = \frac{3!}{(2-1)! (3-2+1)!} (1)^{2-1} (2)^{3-2+1} = 12$
$\displaystyle mq = (2)(2) = 4$