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What is the total number of different 9-letter arrangements that can be formed using the letters in the word PERIMETER? Answer:

What is the total number of different 99-letter arrangements that can be formed using the letters in the word PERIMETER?\newlineAnswer:

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Full solution

Q. What is the total number of different 99-letter arrangements that can be formed using the letters in the word PERIMETER?\newlineAnswer:
  1. Question Prompt: Question_prompt: How many different 99-letter arrangements can be formed from the word PERIMETER?
  2. Count Letters: First, count the number of each letter in PERIMETER. We have P(1)P(1), E(3)E(3), R(2)R(2), I(1)I(1), M(1)M(1), T(1)T(1).
  3. Formula for Arrangements: The formula for the number of arrangements of a word with repeated letters is n!n1!×n2!×…×nk!\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}, where nn is the total number of letters, and n1,n2,…,nkn_1, n_2, \ldots, n_k are the frequencies of the repeated letters.
  4. Calculate Total Factorial: Calculate the factorial of the total number of letters: 9!=9×8×7×6×5×4×3×2×19! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.
  5. Calculate Factorial for Repeated Letters: Calculate the factorial for each of the repeated letters: E(3)E(3) so 3!3!, R(2)R(2) so 2!2!, and the rest are unique so their factorials are 11.
  6. Plug Values into Formula: Now plug these values into the formula: 9!/(3!×2!×1!×1!×1!×1!)=(9×8×7×6×5×4×3×2×1)/((3×2×1)×(2×1))9! / (3! \times 2! \times 1! \times 1! \times 1! \times 1!) = (9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) / ((3 \times 2 \times 1) \times (2 \times 1)).
  7. Simplify Calculation: Simplify the calculation: 9!3!×2!=(9×8×7×6×5×4×3×2×1)(6×2)=(9×8×7×5×4×3×1)1.\frac{9!}{3! \times 2!} = \frac{(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(6 \times 2)} = \frac{(9 \times 8 \times 7 \times 5 \times 4 \times 3 \times 1)}{1}.
  8. Finish Calculation: Finish the calculation: 9×8×7×5×4×3=302409 \times 8 \times 7 \times 5 \times 4 \times 3 = 30240.

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