(8) [15 marks] We are throwing n distinct balls (say of different colors) into n distinct (numbered) bins. Each ball lands in one bin. For example, if n = 2, the number of ways to distribute the 2 different balls (say with colors Red and Green) would be: (Red ball in Bin 1, Green ball in Bin 2), (Green ball in Bin 1, Red ball in Bin 2), (Both Red and Green balls in Bin 1, no balls in Bin 2), (No balls in Bin 1, both Red and Green in Bin 2). e Calculate the number of ways to allocate the n balls among the n bins? [5 marks] e For k > 1, calculate the number of ways to allocate the n balls among the n bins such that bin 1 has exactly k balls? [5 marks| e For k > 1, calculate the number of ways to allocate the n balls among the n bins such that bin 1 has > k balls? [5 marks] (9) [10 marks] First write a combinatorial proof of the following equality ()- 2h). Then use the binomial theorem to write another proof. (Hint: (1+)"-(1+2)" = (1+ 2)?")