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I have $5$ different mathematics textbooks and $4$ different psychology textbooks. In how many ways can I place the $9$ textbooks on a bookshelf, in a row, if all the psychology textbooks must be together, and all the mathematics textbooks must be together?

 Dec 13, 2023
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You have 5 different mathematics textbooks and 4 different psychology textbooks that you want to place on a bookshelf. However, you want to keep all the psychology textbooks together, and all the mathematics textbooks together. This means that you have two groups of textbooks that you need to place on the shelf in a specific order. 

 

To solve this problem, first, you need to determine the number of ways you can arrange the mathematics textbooks and the number of ways you can arrange the psychology textbooks. Since you have 5 different mathematics textbooks, you can arrange them in 5! or 120 different ways. Similarly, you have 4 different psychology textbooks, and you can arrange them in 4! or 24 different ways.

 

Once you have arranged the mathematics and psychology textbooks separately, you need to place them together in a specific order. Since there are only two ways you can group the textbooks (all mathematics textbooks followed by all psychology textbooks or all psychology textbooks followed by all mathematics textbooks), you have only two possible arrangements. 

 

Therefore, the total number of ways you can place all 9 textbooks on the bookshelf, in a row, if all the psychology textbooks must be together, and all the mathematics textbooks must be together, is:

 

2 × 5! × 4! = 2 × 120 × 24 = 5,760

 

So, there are 5,760 ways to place all 9 textbooks on the bookshelf under the given conditions.

 Dec 13, 2023

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