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

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

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Write a linear function: word problems

Full solution

Q. What is the total number of different

10

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

\newline

Answer:

Count Letters: Count the number of each letter in COMMISSION. $\newline$ $C=1$ , $O=2$ , $M=2$ , $I=2$ , $S=2$ , $N=1$ .
Calculate Total Factorial: Calculate the factorial of the total number of letters. $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ .
Divide by Repeating Factorials: Divide by the factorial of the number of times each letter repeats to correct for overcounting. $\newline$ The equation is $10! / (2! \times 2! \times 2! \times 2!)$ .
Do the Math: Do the math: $\frac{10!}{(2! \times 2! \times 2! \times 2!)} = \frac{(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(2 \times 2 \times 2 \times 2)}$ .
Simplify Equation: Simplify the equation: $\frac{10!}{(2! \times 2! \times 2! \times 2!)} = \frac{(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3)}{(2 \times 2 \times 2)}$ .
Calculate Result: Calculate the result: $(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / (2 \times 2 \times 2) = 453600$ .

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