gnatopoditw

2022-06-25

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
$y!=\left(y\right)\left(y-1\right)\left(y-2\right)\dots$
but I have not encountered this before, a fraction inside a factorial. Am I looking for something like
$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)\left(\frac{\left(x-1\right)+\left(y-1\right)}{1}\right),$
and this is where I am stuck. Any help would be great.

tennispopj8

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

Lucille Cummings

The expression is of the form
$\frac{a!\cdot b!}{\left(a-b\right)!},$
provided $x$ and $y$ share the same parity, of course. Now this simplifies (?) to
$\left(\genfrac{}{}{0}{}{a}{b}\right)\cdot b!.$

Do you have a similar question?

Recalculate according to your conditions!