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

What is the total number of different 1111-letter arrangements that can be formed using the letters in the word INTERPOLATE?\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 INTERPOLATE?\newlineAnswer:
  1. Calculate Factorial: The word INTERPOLATE has 1111 letters, with the letter 'T' repeating twice. To find the total number of different 1111-letter arrangements, we need to calculate the factorial of the number of letters and divide by the factorial of the number of times any letter is repeated.
  2. Divide by Repeated Factorial: Calculate the factorial of the total number of letters in the word INTERPOLATE, which is 11!11! (1111 factorial).\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
  3. Divide Factorials: Since the letter 'T' repeats twice, we need to divide the total by the factorial of the number of times 'T' repeats, which is 2!2! (22 factorial).\newline2!=2×12! = 2 \times 1
  4. Simplify Expression: Now, divide 11!11! by 2!2! to get the total number of different arrangements.\newlineTotal arrangements = 11!2!=11×10×9×8×7×6×5×4×3×2×12×1\frac{11!}{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}
  5. Perform Division: Simplify the expression by canceling out the common factors.\newlineTotal arrangements = (11×10×9×8×7×6×5×4×3)/2(11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 2
  6. Perform Division: Simplify the expression by canceling out the common factors.\newlineTotal arrangements = (11×10×9×8×7×6×5×4×3)/2(11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 2 Perform the division to get the final answer.\newlineTotal arrangements = (11×10×9×8×7×6×5×4×3)/2=19958400(11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 2 = 19958400

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