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What is the total number of different 10-letter arrangements that can be formed using the letters in the word SUFFRAGIST? Answer:

What is the total number of different 1010-letter arrangements that can be formed using the letters in the word SUFFRAGIST?\newlineAnswer:

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Full solution

Q. What is the total number of different 1010-letter arrangements that can be formed using the letters in the word SUFFRAGIST?\newlineAnswer:
  1. Count Unique Letters: First, count the number of each unique letter in SUFFRAGIST. \newlineS=2S = 2, U=1U = 1, F=2F = 2, R=1R = 1, A=1A = 1, G=1G = 1, I=1I = 1, T=1T = 1.
  2. Calculate Total Arrangements: Since there are 1010 letters in total, the number of different arrangements without considering repeating letters is 10!10! (1010 factorial).\newlineCalculate 10!=10×9×8×7×6×5×4×3×2×110! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.
  3. Account for Repeating Letters: Now, we need to account for the repeating letters. There are 22 S's and 22 F's.\newlineThe number of arrangements for these repeating letters is 2!2! for each set of repeating letters.\newlineCalculate 2!2! for S's which is 2×12 \times 1 and 2!2! for F's which is also 2×12 \times 1.
  4. Divide Total by Factorials: Divide the total number of arrangements by the product of the factorials of the counts of each repeating letter to correct for overcounting.\newlineSo, divide 10!10! by (2!×2!)(2! \times 2!).
  5. Perform Division: Perform the division: (10!)/(2!×2!)=(10×9×8×7×6×5×4×3×2×1)/((2×1)×(2×1))(10!)/(2! \times 2!) = (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)).
  6. Simplify Calculation: Simplify the calculation: (10×9×8×7×6×5×4×3)/(2×2)=(10×9×8×7×6×5×4×3)/4(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / (2 \times 2) = (10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 4.
  7. Finish Calculation: Finish the calculation: 10×9×8×7×6×5×4×34=10×9×8×7×6×5×2×3\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{4} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 2 \times 3.
  8. Final Answer: The final answer is the product of these numbers: 10×9×8×7×6×5×2×3=181440010 \times 9 \times 8 \times 7 \times 6 \times 5 \times 2 \times 3 = 1814400.

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