In this problem, we investigate the effect of various assumptions on the number of ways of placing n balls into b distinct bins.

a. Suppose that the n balls are distinct and that their order within a bin does not matter. Argue that the number of ways of placing the balls in the bins is bⁿ.

b. Suppose that the balls are distinct and that the balls in each bin are ordered. Prove that there are exactly (b + n – 1)!/(b – 1)! ways to place the balls in the bins. 

c. Suppose that the balls are identical, and hence their order within a bin does not matter. Show that the number of ways of placing the balls in the bins is (^{b + n − 1}_n). Of the arrangements in part (b), how many are repeated if the balls are made identical?

d. Suppose that the balls are identical and that no bin may contain more than one ball, so that n − b. Show that the number of ways of placing the balls is (^b_n).

e. Suppose that the balls are identical and that no bin may be left empty. Assuming that n ≥ b, show that the number of ways of placing the balls is (^n − 1_b – 1).