Consider S = {1, 2, . . . , n}. Alice selects some integers from set S. Call this set A. Bob also selects some integers from set S. Call this set B.
i) How many ways are present of selecting (A, B) such that every number selected by Alice is also selected by Bob?
ii) How many ways are present of selecting (A, B), if every number selected by Alice is also selected by Bob and there is at least one number that Bob selects which is not chosen by Alice?