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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 $10$ -letter arrangements that can be formed using the letters in the word SUFFRAGIST? $\newline$ Answer:

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Convert between customary and metric systems

Full solution

Q. What is the total number of different

10

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

\newline

Answer:

Count Unique Letters: First, count the number of each unique letter in SUFFRAGIST. $\newline$ $S = 2$ , $U = 1$ , $F = 2$ , $R = 1$ , $A = 1$ , $G = 1$ , $I = 1$ , $T = 1$ .
Calculate Total Arrangements: Since there are $10$ letters in total, the number of different arrangements without considering repeating letters is $10!$ ( $10$ factorial). $\newline$ Calculate $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ .
Account for Repeating Letters: Now, we need to account for the repeating letters. There are $2$ S's and $2$ F's. $\newline$ The number of arrangements for these repeating letters is $2!$ for each set of repeating letters. $\newline$ Calculate $2!$ for S's which is $2 \times 1$ and $2!$ for F's which is also $2 \times 1$ .
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. $\newline$ So, divide $10!$ by $(2! \times 2!)$ .
Perform Division: Perform the division: $(10!)/(2! \times 2!) = (10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) / ((2 \times 1) \times (2 \times 1))$ .
Simplify Calculation: Simplify the calculation: $(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / (2 \times 2) = (10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 4$ .
Finish Calculation: Finish the calculation: $\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{4} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 2 \times 3$ .
Final Answer: The final answer is the product of these numbers: $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 2 \times 3 = 1814400$ .

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