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What is the total number of different 8-letter arrangements that can be formed using the letters in the word ZIGGURAT? Answer:

What is the total number of different 88-letter arrangements that can be formed using the letters in the word ZIGGURAT?\newlineAnswer:

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Full solution

Q. What is the total number of different 88-letter arrangements that can be formed using the letters in the word ZIGGURAT?\newlineAnswer:
  1. Identify Unique Letters: Identify the unique letters and their frequencies in the word ZIGGURAT.\newlineZIGGURAT contains the letters ZZ, II, GG, UU, RR, AA, TT. The letter GG appears twice, while all other letters appear once.
  2. Calculate Total Arrangements: Calculate the total number of arrangements without considering the repetition of the letter G.\newlineThe total number of arrangements of 88 letters is 8!8! (88 factorial), which is 8×7×6×5×4×3×2×18 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.
  3. Calculate 8!8!: Calculate 8!8! to find the total number of arrangements without considering the repetition.\newline8!=8×7×6×5×4×3×2×1=40,3208! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320
  4. Adjust for Repetition: Adjust for the repetition of the letter G. Since the letter G is repeated twice, we need to divide the total number of arrangements by the number of arrangements of the repeated letters to avoid overcounting. The number of arrangements of 22 Gs is 2!2! (22 factorial), which is 2×12 \times 1.
  5. Calculate 2!2!: Calculate 2!2! to find the number of arrangements of the repeated Gs.\newline2!=2×1=22! = 2 \times 1 = 2
  6. Divide Total Arrangements: Divide the total number of arrangements by the number of arrangements of the repeated Gs to find the correct total number of different arrangements.\newlineThe correct total number of different 88-letter arrangements is 40,320÷240,320 \div 2.
  7. Evaluate Final Answer: Evaluate 40,320÷240,320 \div 2 to find the final answer.\newline40,320÷2=20,16040,320 \div 2 = 20,160

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