Factorial simplification with fractions

I am attempting to simplify the expression

$\frac{\left(\frac{x+y}{2}\right)!\left(\frac{x-y}{2}\right)!}{y!}$

I'm familiar with expanding expressions like

$y!=(y)(y-1)(y-2)\dots $

but I have not encountered this before, a fraction inside a factorial. Am I looking for something like

$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)\left(\frac{(x-1)+(y-1)}{1}\right),$

and this is where I am stuck. Any help would be great.

I am attempting to simplify the expression

$\frac{\left(\frac{x+y}{2}\right)!\left(\frac{x-y}{2}\right)!}{y!}$

I'm familiar with expanding expressions like

$y!=(y)(y-1)(y-2)\dots $

but I have not encountered this before, a fraction inside a factorial. Am I looking for something like

$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)\left(\frac{(x-1)+(y-1)}{1}\right),$

and this is where I am stuck. Any help would be great.