Factorial simplification with fractions

Question

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

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.

Answer 1

$$\left( \frac{x+y}{2} \right)!=\left( \frac{x+y}{2} \right)\left( \frac{x+y}{2} -1 \right)\left( \frac{x+y}{2} -2 \right)\cdots 3\cdot 2\cdot 1$$

Your original expression is about as simplified as it's going to get. It is only defined if $x,y$ are both even, or both odd.

Answer 2

The expression is of the form $$\frac{a!\cdot b!}{(a-b)!},$$ provided $x$ and $y$ share the same parity, of course. Now this simplifies (?) to $$\binom{a}{b}\cdot b!.$$