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

What is the total number of different 1111-letter arrangements that can be formed using the letters in the word PRECIPITOUS?\newlineAnswer:

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Full solution

Q. What is the total number of different 1111-letter arrangements that can be formed using the letters in the word PRECIPITOUS?\newlineAnswer:
  1. Count Letters: First, count the number of each letter in the word PRECIPITOUS. PP appears 22 times, RR appears 11 time, EE appears 11 time, CC appears 11 time, II appears 22 times, 2200 appears 11 time, 2222 appears 11 time, 2244 appears 11 time, 2266 appears 11 time.
  2. Formula for Arrangements: The formula for the number of arrangements of a word with repeated letters is n!/(p1!∗p2!∗...∗pk!)n! / (p_1! * p_2! * ... * p_k!), where nn is the total number of letters, and p1,p2,...,pkp_1, p_2, ..., p_k are the frequencies of the repeated letters.
  3. Calculate Total Factorial: Calculate the factorial of the total number of letters, which is 11!11! for PRECIPITOUS.\newline11!=11×10×9×8×7×6×5×4×3×2×111! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
  4. Calculate Factorial for Repeated Letters: Calculate the factorial for the repeated letters PP and II, which appear 22 times each.\newline2!=2×12! = 2 \times 1
  5. Divide Factorials: Now, divide 11!11! by the product of the factorials of the frequencies of the repeated letters.\newlineSo, the number of different arrangements is 11!(2!⋅2!)\frac{11!}{(2! \cdot 2!)}.
  6. Perform Calculation: Perform the calculation: 11!/(2!×2!)=11×10×9×8×7×6×5×4×3×2×12×1×2×111! / (2! \times 2!) = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1}
  7. Simplify Calculation: Simplify the calculation: 11!(2!×2!)=(11×10×9×8×7×6×5×4×3)(2×2)\frac{11!}{(2! \times 2!)} = \frac{(11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3)}{(2 \times 2)}
  8. Finish Calculation: Finish the calculation: 11!/(2!×2!)=(11×10×9×8×7×6×5×4×3)/411! / (2! \times 2!) = (11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 4
  9. Final Answer: The final answer is 11!/(2!×2!)=11×10×9×8×7×6×5×4×3/4=997920011! / (2! \times 2!) = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 / 4 = 9979200.

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