Problem 142. (Relevant in Appendix C.) The total number of partitions of a k-element set is denoted by B(k) and is called the k-th Bell number. Thus B(1) = 1 and B(2) = 2.
(a) Show, by explicitly exhibiting the partitions, that B(3) = 5.
(b) Find a recurrence that expresses B(k) in terms of B(n) for n < k and
prove your formula correct in as many ways as you can. (h) (c) FindB(k)fork=4,5,6.