Breaking apart factorials

Question

Say you have a factorial like this in an equality:

$\frac{(x-1)!}{((x-1)!-(y-1)!)!}$

Is there any way to split it apart? How can it be manipulated? The second factorial seems to complicate things.

Answer 1

$$\frac{(x-1)!}{((x-1)!-(y-1)!)!}=\frac{(x-1)!}{\left(-\frac{xy!-yx!}{xy}\right)!}=\frac{\Gamma(x)}{\Gamma\left(\Gamma(x)-\Gamma(y)-1\right)}=\frac{\Gamma(x)}{\left(\Gamma(x)-\Gamma(y)\right)!}$$