probability

Question

Four books are put on two shelves so that none of the shelves are empty. In how many
ways can it be done if both books and shelves can be distinguishable and indistinguishable?
Consider all four cases.

Answer 1

If all books are distinctly identifiable, and if the arrangements on the shelf are also of importance (e.g. AB and BA, ordered from left to right, are also significant, then

Top shelf may contain 4P1, 4P2 or 4P3 while concurrently(by default) bottom shelf would be 3P3, 2P2 and 1P1, respectively.

Thus the top shelf can be done 4 ways, or 12 ways or 24 ways. These top-shelf arrangements are paired with 6, 2 or 1 ways, respectively. The three products of these respective numbers are each 24; my answer is 72.

 

I will leave the other case(s) for you to play with.