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What is the total number of different 12-letter arrangements that can be formed using the letters in the word CIVILIZATION? Answer:

What is the total number of different 1212-letter arrangements that can be formed using the letters in the word CIVILIZATION?\newlineAnswer:

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Full solution

Q. What is the total number of different 1212-letter arrangements that can be formed using the letters in the word CIVILIZATION?\newlineAnswer:
  1. Count Letters and Frequencies: Determine the total number of letters and the frequency of each letter in the word CIVILIZATION.\newlineCIVILIZATION has 1212 letters in total.\newlineThe frequency of each letter is as follows:\newlineC−1C - 1, I−4I - 4, V−1V - 1, L−1L - 1, Z−1Z - 1, A−1A - 1, T−1T - 1, O−1O - 1, N−1N - 1.
  2. Calculate Total Arrangements: Calculate the total number of arrangements using the formula for permutations of a multiset:\newlineThe formula is n!/(n1!∗n2!∗…∗nk!)n! / (n_1! * n_2! * \ldots * n_k!), where nn is the total number of items to arrange, and n1,n2,…,nkn_1, n_2, \ldots, n_k are the frequencies of each distinct item.\newlineFor CIVILIZATION, this is 12!/(1!∗4!∗1!∗1!∗1!∗1!∗1!∗1!∗1!)12! / (1! * 4! * 1! * 1! * 1! * 1! * 1! * 1! * 1!).
  3. Evaluate Factorial Expressions: Evaluate the factorial expressions and perform the division.\newline12!=479,001,60012! = 479,001,600 (This is the number of ways to arrange 1212 distinct letters.)\newline4!=244! = 24 (This is the number of ways to arrange the four I's among themselves.)\newlineSo, the total number of different arrangements is 479,001,60024\frac{479,001,600}{24}.
  4. Find Final Answer: Calculate the division to find the final answer. 479,001,600/24=19,958,400479,001,600 / 24 = 19,958,400.

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