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What is the total number of different 13 -letter arrangements that can be formed using the letters in the word REVERBERATION? Answer:

What is the total number of different 1313 -letter arrangements that can be formed using the letters in the word REVERBERATION?\newlineAnswer:

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Full solution

Q. What is the total number of different 1313 -letter arrangements that can be formed using the letters in the word REVERBERATION?\newlineAnswer:
  1. Count and Identify Letters: Count the letters in REVERBERATION and identify any repeating letters.\newlineREVERBERATION has 1313 letters with the following counts: R−3R-3, E−3E-3, V−1V-1, B−1B-1, A−2A-2, T−1T-1, I−1I-1, O−1O-1, N−1N-1.
  2. Calculate Total Arrangements: Calculate the factorial of the total number of letters to find the total arrangements without considering repetitions.\newlineTotal arrangements = 13!=13×12×11×…×113! = 13 \times 12 \times 11 \times \ldots \times 1.
  3. Calculate Factorials for Repeating Letters: Calculate the factorial of the number of each repeating letter to account for indistinguishable arrangements.\newlineR has 33 repeats, E has 33 repeats, and A has 22 repeats.\newlineSo, we have 3!3! for R, 3!3! for E, and 2!2! for A.
  4. Divide Total Arrangements: Divide the total arrangements by the product of the factorials of the repeating letters to get the number of distinct arrangements.\newlineDistinct arrangements = 13!(3!×3!×2!)\frac{13!}{(3! \times 3! \times 2!)}.
  5. Perform Calculations: Perform the calculations.\newline13!=6,227,020,80013! = 6,227,020,800\newline3!=63! = 6\newline2!=22! = 2\newlineDistinct arrangements = 6,227,020,800(6×6×2)=6,227,020,80072=86,486,400\frac{6,227,020,800}{(6 \times 6 \times 2)} = \frac{6,227,020,800}{72} = 86,486,400.

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