1. **Problem Statement:** Find the number of distinct permutations of the word "MATHEMATICS" considering repeated letters. 2. **Understanding the problem:** The word "MATHEMATICS" has 11 letters in total. - The letter M appears 2 times. - The letter A appears 2 times. - The letter T appears 2 times. 3. **Formula for permutations with repeated letters:** $$\text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of letters, and $n_1, n_2, ..., n_k$ are the counts of each repeated letter. 4. **Applying the formula:** $$\frac{11!}{2! \times 2! \times 2!}$$ 5. **Calculating factorial values:** - $11! = 39916800$ - $2! = 2$ 6. **Simplify the denominator:** $$2! \times 2! \times 2! = 2 \times 2 \times 2 = 8$$ 7. **Calculate the total permutations:** $$\frac{39916800}{8} = 4989600$$ 8. **Second expression given:** $$\frac{9!}{2! \times 2!} \times \frac{4!}{2!}$$ 9. **Calculate factorials:** - $9! = 362880$ - $4! = 24$ - $2! = 2$ 10. **Simplify each fraction:** $$\frac{9!}{2! \times 2!} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720$$ $$\frac{4!}{2!} = \frac{24}{2} = 12$$ 11. **Multiply the two results:** $$90720 \times 12 = 1088640$$ 12. **Interpretation:** - The first expression gives the total distinct permutations of "MATHEMATICS" considering repeated letters. - The second expression is a product of permutations of subsets of letters, possibly representing a different counting approach. **Final answers:** - Total distinct permutations of "MATHEMATICS": $4989600$ - Value of the product expression: $1088640$