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

What is the total number of different 1313 -letter arrangements that can be formed using the letters in the word SCARIFICATION?\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 SCARIFICATION?\newlineAnswer:
  1. Count Letters: Count the number of each letter in SCARIFICATION: S=1S=1, C=2C=2, A=2A=2, R=1R=1, I=3I=3, F=1F=1, O=1O=1, T=1T=1, N=1N=1.
  2. Calculate Factorial: Calculate the factorial of the total number of letters, which is 13!13! for the different arrangements if all letters were unique.\newline13!=13×12×11×10×9×8×7×6×5×4×3×2×113! = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.
  3. Divide Total Arrangements: Divide the total arrangements by the factorial of the number of times each letter repeats to correct for overcounting.\newlineSo we divide by 2!2! for C, 2!2! for A, and 3!3! for I.\newlineThe correct expression is 13!(2!×2!×3!)\frac{13!}{(2! \times 2! \times 3!)}.
  4. Calculate Factorials: Calculate the factorials: 2!=22! = 2 and 3!=63! = 6.
  5. Plug in Values: Now, plug in the values and calculate the expression: 13!/(2!×2!×3!)=(13×12×11×10×9×8×7×6×5×4×3×2×1)/(2×2×6)13! / (2! \times 2! \times 3!) = (13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) / (2 \times 2 \times 6).
  6. Simplify Expression: Simplify the expression by canceling out common factors.\newlineThe expression simplifies to (13×12×11×10×9×8×7×5×4×3×1)/(1×1×1)(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 5 \times 4 \times 3 \times 1) / (1 \times 1 \times 1).
  7. Calculate Simplified Expression: Calculate the simplified expression: 13×12×11×10×9×8×7×5×4×3×1=1,235,52013 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 5 \times 4 \times 3 \times 1 = 1,235,520.

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