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Chapter 1: Problem 15

In how many ways can Beth place 24 different books on four shelves so that there is at least one book on each shelf? (For any of these arrangements consider the books on each shelf to be placed one next to the other, with the first book at the left of the shelf.)

Short Answer

Expert verified
The total number of arrangements will be the product of the two steps' results, i.e., \(24 \times 23 \times 22 \times 21 \times 4^{20}\)

Step by step solution

01

Distribute one book to each shelf

To ensure each shelf gets at least one book, pick one book for each shelf. There are \(24 \times 23 \times 22 \times 21\) ways of doing this as there are 24 choices for the first shelf, 23 for the second, 22 for the third, and 21 for the fourth.
02

Distribute the remaining books

After giving one book to each shelf, we have 20 books left. Each book can now go to any of the 4 shelves. So there are \(4^{20}\) ways of doing this.
03

Consider the order of books on each shelf

The books on each shelf can be arranged in any order from left to right. If there are \(n\) books on a shelf, there are \(n!\) ways to arrange them. However, as we do not know exactly how many books end up on each shelf, this possibility is already implicitly taken into account in the preceding steps.

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