Binomial expansion + interesting question

Hey,

how would I use a 'combinatoral proof' to prove this:

http://upload.wikimedia.org/math/a/6...a31629e599.png

and here's a question that has got me puzzled..

A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates.

(a) If the pairing is done at random, what is the probability that there are no offensive-defensive pairs?

(b) What is the probability that there are 2i offensive-defensive roommate pairs (with i= 1,2,...,10)

Thanks in advance