What is the total number of different 9-letter arrangements that can be formed using the letters in the word AMENDMENT? Answer:

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Count Letters and Frequencies: Determine the total number of letters and the frequency of each letter in the word AMENDMENT. The word AMENDMENT has 9 letters in total. The letter frequencies are as follows: A - 1 time M - 2 times E - 2 times N - 2 times D - 1 time T - 1 timeUse Permutations Formula: Use the formula for permutations of a multiset to calculate the number of different arrangements. The formula is: Number of arrangements = n! / (n1! * n2! * ... * nk!) where n is the total number of letters, and n1, n2, ..., nk are the frequencies of each distinct letter. For AMENDMENT, this becomes: Number of arrangements = 9! / (1! * 2! * 2! * 2! * 1! * 1!)Calculate Total Factorial: Calculate the factorial of the total number of letters, which is 9!. 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880Calculate Letter Factorials: Calculate the factorial for each of the letter frequencies. Since the frequencies are 1 or 2, we only need to calculate 2!. 2! = 2 * 1 = 2 1! = 1 (but we don't really need to calculate this since the factorial of 1 is 1)Divide Factorials for Arrangements: Divide the total factorial by the product of the factorials of the letter frequencies. Number of arrangements = 362,880 / (1 * 2 * 2 * 2 * 1 * 1) Number of arrangements = 362,880 / 8 Number of arrangements = 45,360

What is the total number of different $9$ -letter arrangements that can be formed using the letters in the word AMENDMENT? $\newline$ Answer:

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