1. **Problem:** Find the number of ways to arrange the letters of the word \textbf{MATHEMATICS}. 2. **Understanding the problem:** The word MATHEMATICS has 11 letters with some letters repeating. The formula for the number of arrangements of $ n $ letters where some letters repeat is: $$\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \cdots}$$ where $ n $ is the total number of letters, and $ n_1, n_2, \ldots $ are the frequencies of each repeated letter. 3. **Count the letters:** - M appears 2 times - A appears 2 times - T appears 2 times - H appears 1 time - E appears 1 time - I appears 1 time - C appears 1 time - S appears 1 time Total letters $ n = 11 $. 4. **Apply the formula:** $$\text{Number of arrangements} = \frac{11!}{2! \times 2! \times 2!}$$ 5. **Calculate factorials:** $$11! = 39916800$$ $$2! = 2$$ 6. **Simplify denominator:** $$2! \times 2! \times 2! = 2 \times 2 \times 2 = 8$$ 7. **Final calculation:** $$\frac{39916800}{8} = 4989600$$ **Answer:** There are $4989600$ ways to arrange the letters of the word MATHEMATICS.