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What is the total number of different 6-letter arrangements that can be formed using the letters in the word QUASAR?
Answer:

What is the total number of different $6$ -letter arrangements that can be formed using the letters in the word QUASAR? $\newline$ Answer:

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Convert between customary and metric systems

Full solution

Q. What is the total number of different

6

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

\newline

Answer:

Count Letters and Repeats: Determine the number of letters in the word QUASAR and identify any repeating letters. $\newline$ The word QUASAR has $6$ letters, with the letter $A$ repeating twice.
Calculate Total Arrangements: Calculate the total number of arrangements using the formula for permutations of $n$ items with repetitions: $\newline$ The formula is $\frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}$ , where $n$ is the total number of items, and $p_1, p_2, \ldots, p_k$ are the numbers of identical items. $\newline$ For QUASAR, $n = 6$ (total letters), and there is one letter (A) that repeats twice, so $p_1 = 2$ . $\newline$ The permutation formula for QUASAR is $\frac{6!}{2!}$ .
Factorial of Total Letters: Calculate the factorial of the total number of letters ( $6!$ ). $\newline$ $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
Factorial of Repeating Letters: Calculate the factorial of the number of repeating letters $2!$ . $2! = 2 \times 1 = 2$
Divide Factorials for Arrangements: Divide the factorial of the total number of letters by the factorial of the number of repeating letters to find the total number of different arrangements. $\newline$ Total arrangements = $\frac{6!}{2!} = \frac{720}{2} = 360$

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