1. **Problem statement:** We need to find the number of ways to divide 17 chapters among four writers such that the first and third writers write 5 chapters each, the second writer writes 4 chapters, and the fourth writer writes 3 chapters. 2. **Understanding the problem:** The total chapters are 17, and the distribution is fixed: 5 chapters to writer 1, 4 chapters to writer 2, 5 chapters to writer 3, and 3 chapters to writer 4. 3. **Formula used:** The number of ways to divide distinct items into groups of fixed sizes is given by the multinomial coefficient: $$\frac{17!}{5! \times 4! \times 5! \times 3!}$$ 4. **Explanation:** - First, choose 5 chapters out of 17 for writer 1: $\binom{17}{5}$ ways. - Then choose 4 chapters out of the remaining 12 for writer 2: $\binom{12}{4}$ ways. - Then choose 5 chapters out of the remaining 8 for writer 3: $\binom{8}{5}$ ways. - The remaining 3 chapters go to writer 4 automatically. 5. **Intermediate calculation:** $$\binom{17}{5} \times \binom{12}{4} \times \binom{8}{5} = \frac{17!}{5! \times 12!} \times \frac{12!}{4! \times 8!} \times \frac{8!}{5! \times 3!}$$ 6. **Simplify by canceling factorials:** $$= \frac{17!}{\cancel{5!} \times \cancel{12!}} \times \frac{\cancel{12!}}{\cancel{4!} \times \cancel{8!}} \times \frac{\cancel{8!}}{\cancel{5!} \times 3!} = \frac{17!}{5! \times 4! \times 5! \times 3!}$$ 7. **Final answer:** The number of ways to divide the book is $$\boxed{\frac{17!}{5! \times 4! \times 5! \times 3!}}$$