If we take 2 items from 4 distinct items then that occurs in 4x3=12 ways, which is just 
now to pair these items off we divide by 2x2 as we are taking 2 items and we want to form pairs with these. therefore we get that
. The 2x2 represents that we are pairing off the 4 objects into 2's, so we first divide by 2 to form pairs, and then divide by 2 again to ensure distinct pairs.
Now apply this logic for
. We take n objects from 2n objects in 
ways. Now divide by 2 to form pairs, and keep dividing by 2 a further (n-1) times to form distinct pair offs to obtain:
!}{2^{n}(n!)})
now to pair these items off we divide by 2x2 as we are taking 2 items and we want to form pairs with these. therefore we get that
Now apply this logic for