Step-by-step Solution:

Given: There are: - 4 books on fairy tales, - 5 novels, and - 3 plays. These books are arranged in a specific order: books on fairy tales, novels, and then plays. The 4 books on fairy tales must be kept together. They can be arranged among themselves in: \[ 4! = 24 \] The 5 novels can be arranged among themselves in: \[ 5! = 120 \] The 3 plays can be arranged among themselves in: \[ 3! = 6 \] Since the 3 categories (fairy tales, novels, and plays) are distinct groups, they can be arranged in: \[ 3! = 6 \] Total Number of Arrangements: Using the counting principle, the total number of ways to arrange all books is: \[ 3! \times (4! \times 5! \times 3!) \] \[ = 6 \times (24 \times 120 \times 6) \] \[ = 6 \times 17280 \] \[ = 103680 \] Thus, the total number of ways to arrange the books is 103680.