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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 $8$ -letter arrangements that can be formed using the letters in the word ZIGGURAT? $\newline$ Answer:

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Convert between customary and metric systems

Full solution

Q. What is the total number of different

8

-letter arrangements that can be formed using the letters in the word ZIGGURAT?

\newline

Answer:

Identify Unique Letters: Identify the unique letters and their frequencies in the word ZIGGURAT. $\newline$ ZIGGURAT contains the letters $Z$ , $I$ , $G$ , $U$ , $R$ , $A$ , $T$ . The letter $G$ appears twice, while all other letters appear once.
Calculate Total Arrangements: Calculate the total number of arrangements without considering the repetition of the letter G. $\newline$ The total number of arrangements of $8$ letters is $8!$ ( $8$ factorial), which is $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ .
Calculate $8!$ : Calculate $8!$ to find the total number of arrangements without considering the repetition. $\newline$ $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320$
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 $2$ Gs is $2!$ ( $2$ factorial), which is $2 \times 1$ .
Calculate $2!$ : Calculate $2!$ to find the number of arrangements of the repeated Gs. $\newline$ $2! = 2 \times 1 = 2$
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. $\newline$ The correct total number of different $8$ -letter arrangements is $40,320 \div 2$ .
Evaluate Final Answer: Evaluate $40,320 \div 2$ to find the final answer. $\newline$ $40,320 \div 2 = 20,160$

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