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 \(\displaystyle 14! \times 1!\). Now, if there are 13 books on Shelf A and 2 are on Shelf B, the number of possible arrangements are \(\displaystyle 13! \times 2!\). And so it goes on as such and so the total possible arrangements would be \(\displaystyle 2(14! \times 1! + 13!\times 2! + ... + 9! \times 6! + 8! \times 7!)\). Is this correct or am I just way off? Thanks.