i dont know how sound this logic is since i haven't done factorials in a long time but maybe it will steer you in some sort of direction
(2n)!
------
(2n-2)!
gives:
1x2x3...x(2n-3) x(2n-2) x (2n-1) x (2n)
--------------------------------------
1x2x3...x((2n-2)-1) x (2n-2)
this leaves (2n-1)(2n)