HINT for Problem 3 Extension
It takes some thought just to interpret the left-hand side. Given a collection of n balls, then there are C(n,2) pairs of balls available. Now imagine choosing two of these pairs. Be careful; it's possible that the two pairs could have a ball in common! Now imagine including an extra hot pink ball with the others, and establish a correspondence between choosing two pairs from the original n balls and choosing four balls from the new set of n+1 balls.