1. **Stating the problem:** We want to arrange 4 textbooks, 3 exercise books, and 2 manuals on a shelf. We need to find the number of ways to do this under different conditions: a) No restrictions b) A textbook must be first c) Exercise books must be together d) Textbooks, exercise books, and manuals each must be grouped together 2. **Formulas and rules:** - The number of ways to arrange $n$ distinct items is $n!$. - If some items are grouped, treat the group as a single item, then multiply by the arrangements inside the group. 3. **Solution:** a) No restrictions: Total items = $4 + 3 + 2 = 9$ Number of ways = $9!$ b) Textbook first: Fix one textbook at first position: $4$ choices Arrange remaining $8$ items: $8!$ Number of ways = $4 \times 8!$ c) Exercise books together: Treat 3 exercise books as one group. Number of items to arrange = $4$ textbooks + $1$ group + $2$ manuals = $7$ Arrange these: $7!$ Arrange exercise books inside group: $3!$ Number of ways = $7! \times 3!$ d) Textbooks, exercise books, and manuals each together: Treat each group as one item: 3 groups Arrange groups: $3!$ Arrange inside each group: - Textbooks: $4!$ - Exercise books: $3!$ - Manuals: $2!$ Number of ways = $3! \times 4! \times 3! \times 2!$ 4. **Final answers:** - a) $9! = 362880$ - b) $4 \times 8! = 4 \times 40320 = 161280$ - c) $7! \times 3! = 5040 \times 6 = 30240$ - d) $3! \times 4! \times 3! \times 2! = 6 \times 24 \times 6 \times 2 = 1728$