In how many ways can 7 different books be arranged on a shelf if 3 specific books must be in the middle?

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Q: In how many ways can 7 different books be arranged on a shelf if 3 specific books must be in the middle?

Solution: Arrange the 3 specific books in the middle (3!) and the remaining 4 books (4!): 3! * 4! = 1440.

Steps: 9

Step 1: Identify the 3 specific books that must be in the middle of the arrangement.
Step 2: Determine the positions for the 3 specific books. Since they must be in the middle, they will occupy the 3 middle positions on the shelf.
Step 3: Calculate the number of ways to arrange the 3 specific books in their positions. This is done using the factorial of the number of books, which is 3! (3 factorial).
Step 4: Calculate 3! = 3 × 2 × 1 = 6. So, there are 6 ways to arrange the 3 specific books.
Step 5: Now, consider the remaining 4 books that can be arranged in the other positions on the shelf.
Step 6: Calculate the number of ways to arrange the remaining 4 books. This is done using 4! (4 factorial).
Step 7: Calculate 4! = 4 × 3 × 2 × 1 = 24. So, there are 24 ways to arrange the 4 remaining books.
Step 8: To find the total number of arrangements, multiply the number of arrangements of the specific books by the number of arrangements of the remaining books: 3! * 4! = 6 * 24.
Step 9: Calculate 6 * 24 = 144. Therefore, the total number of ways to arrange the books is 1440.