1. **Problem:** In how many ways can you arrange 5 Mathematics books, 4 Science books, and 3 English books on a shelf such that books of the same subject are kept together? 2. **Formula and Explanation:** - Since books of the same subject must be together, treat each subject group as a single item first. - Number of ways to arrange the 3 groups (Mathematics, Science, English) is $3!$. - Within each group, arrange the books: - Mathematics books: $5!$ - Science books: $4!$ - English books: $3!$ - Total arrangements = $3! \times 5! \times 4! \times 3!$ 3. **Calculation:** - $3! = 6$ - $5! = 120$ - $4! = 24$ - $3! = 6$ - Total = $6 \times 120 \times 24 \times 6 = 103680$ **Final answer:** There are $103680$ ways to arrange the books with each subject's books together.