What is the total number of different 11-letter arrangements that can be formed using the letters in the word RESPIRATION? Answer:

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Count Letters Frequency: Determine the total number of letters and the frequency of each letter in the word RESPIRATION. The word RESPIRATION has 11 letters in total. The frequency of each letter is as follows: R - 2 times E - 1 time S - 1 time P - 1 time I - 2 times A - 1 time T - 1 time O - 1 time N - 1 timePermutations 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 RESPIRATION, this becomes: Number of arrangements = 11! / (2! * 1! * 1! * 1! * 2! * 1! * 1! * 1! * 1!)Calculate Total Factorial: Calculate the factorial of the total number of letters, which is 11!. 11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1Calculate Letter Factorials: Calculate the factorial of the frequency of each letter. 2! = 2 * 1 1! = 1 (for all the letters that appear only once)Substitute and Calculate: Substitute the factorial values into the formula and calculate the number of arrangements. Number of arrangements = 11! / (2! * 1! * 1! * 1! * 2! * 1! * 1! * 1! * 1!) = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * 1 * 1 * 1 * (2 * 1) * 1 * 1 * 1 * 1) = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 2) = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 1) / 4 = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 / 4 = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 3 = 11 * 10 * 9 * 8 * 7 * 6 * 15 = 11 * 10 * 9 * 8 * 7 * 90 = 11 * 10 * 9 * 8 * 630 = 11 * 10 * 9 * 5040 = 11 * 10 * 45360 = 11 * 453600 = 4989600

What is the total number of different $11$ -letter arrangements that can be formed using the letters in the word RESPIRATION? $\newline$ Answer:

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