To determine the number of ways to arrange the letters in the word "MATHEMATICS," we need to consider the total number of letters and the repetitions of certain letters. Let's break this down step-by-step. ### Step 1: Count the Total Letters The word "MATHEMATICS" consists of 11 letters in total. ### Step 2: Identify Repeated Letters Next, we need to identify how many times each letter appears in the word: - M: 2 times - A: 2 times - T: 2 times - H: 1 time - E: 1 time - I: 1 time - C: 1 time - S: 1 time ### Step 3: Use the Formula for Arrangements When arranging letters where some letters are repeated, the formula to find the number of distinct arrangements is given by: \[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \] Where: - \( n \) is the total number of letters, - \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters. In our case: - \( n = 11 \) (total letters) - The frequencies of the repeated letters are: - M: 2 - A: 2 - T: 2 Thus, we can substitute these values into the formula: \[ \text{Number of arrangements} = \frac{11!}{2! \times 2! \times 2!} \] ### Step 4: Calculate the Factorials Now, let's calculate the factorials: - \( 11! = 39916800 \) - \( 2! = 2 \) So, we have: \[ \text{Number of arrangements} = \frac{39916800}{2 \times 2 \times 2} = \frac{39916800}{8} = 4989600 \] ### Step 5: Analyze the Options Now, let's look at the options provided: - **A. \( \frac{11!}{9! 2!} \)**: This option is incorrect because it does not account for the three letters (M, A, T) that are repeated twice. - **B. \( \frac{11!}{9! 2! 2!} \)**: This option is also incorrect because it only accounts for two letters being repeated, not three. - **C. \( \frac{11!}{2! 2! 2!} \)**: This is the correct option as it accurately reflects the repetitions of M, A, and T. - **D. \( \frac{11!}{2! 2!} \)**: This option is incorrect because it does not account for the third letter (T) that is repeated. ### Conclusion The correct answer is **C. \( \frac{11!}{2! 2! 2!} \)**, which gives us the total number of distinct arrangements of the letters in "MATHEMATICS." ### Revision Summary - The total number of letters in "MATHEMATICS" is 11. - The letters M, A, and T each appear twice. - The formula for arrangements with repetitions is \( \frac{n!}{p_1! \times p_2! \times \ldots} \). - The correct calculation for the arrangements is \( \frac{11!}{2! 2! 2!} \).