I know that $\frac{(m-1)!}{(m-n)!(n-1)!} + \frac{(m-1)!}{(m-n-1)!(n)!} = \frac{m!}{(n)!(m-n)!}$, but I am not sure on the intermediate steps. The only solution I am seeing involves finding a common denominator:
$\frac{(m-1)!}{(m-n)!(n-1)!} + \frac{(m-1)!}{(m-n-1)!(n)!} = \frac{(m-1)!(m-n-1)!(n)!+(m-1)!(m-n)!(n-1)!}{(m-n)!(n-1)!(m-n-1)!(n)!}$
But I don't really see where to go from there. Is there a more simple way to go about this? If not, how do I simplify the monster expression above?