1. **Problem statement:** We want to find the number of ways to arrange 7 different books on a shelf such that two particular books are placed at the ends. 2. **Understanding the problem:** There are 7 distinct books, and two specific books must be at the two ends of the shelf. 3. **Step 1: Place the two particular books at the ends.** - There are 2 particular books and 2 ends. - These two books can be arranged in $2! = 2$ ways (one at the left end and the other at the right end, or vice versa). 4. **Step 2: Arrange the remaining books.** - After placing the two particular books, there are $7 - 2 = 5$ books left. - These 5 books can be arranged in any order in the 5 middle positions. - Number of ways to arrange these 5 books is $5! = 120$. 5. **Step 3: Calculate total arrangements.** - Total number of arrangements = ways to arrange the two particular books at ends $\times$ ways to arrange the remaining 5 books - $$2! \times 5! = 2 \times 120 = 240$$ 6. **Final answer:** There are **240** ways to arrange the 7 books on the shelf with the two particular books at the ends.