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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?

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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.

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