You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.)
I want to say $7^5$ since there are 7 possible shelves and five different options to select from books that will be placed.
Could some one explain why my answer is wrong, and what the right answer is?
Suppose you chose Bookshelf 1 five times. That would be just one of the possibilities you've enumerated. But there are 5! = 120 ways to arrange those 5 books on the first bookshelf.
One way to do it: Consider the different shelves as 6 "boundaries" to insert between your books. Thus there are 11 things to order, but 6 of them are equivalent. So there are $\frac{11!}{6!}$ ways.