Not really probability/statistics but this is an introductory discrete mathematics class and I'm rusty on all of this material, so here it goes:

Pamela has 15 different books. In how many ways can she place her books on two shelves so that there is at least one book on each shelf?

So I considered the one event where 14 books are on Shelf A and only 1 on Shelf B, and so the possible arrangement of books in that scenario is

. Now, if there are 13 books on Shelf A and 2 are on Shelf B, the number of possible arrangements are

. And so it goes on as such and so the total possible arrangements would be

. Is this correct or am I just way off? Thanks.