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What is the total number of different 9-letter arrangements that can be formed using the letters in the word AMENDMENT?
Answer:

What is the total number of different $9$ -letter arrangements that can be formed using the letters in the word AMENDMENT? $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. What is the total number of different

9

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

\newline

Answer:

Count Letters and Frequencies: Determine the total number of letters and the frequency of each letter in the word AMENDMENT. $\newline$ The word AMENDMENT has $9$ letters in total. The letter frequencies are as follows: $\newline$ A - $1$ time $\newline$ M - $2$ times $\newline$ E - $2$ times $\newline$ N - $2$ times $\newline$ D - $1$ time $\newline$ T - $1$ time
Use Permutations Formula: Use the formula for permutations of a multiset to calculate the number of different arrangements. $\newline$ The formula is: $\newline$ Number of arrangements = $\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}$ $\newline$ where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the frequencies of each distinct letter. $\newline$ For AMENDMENT, this becomes: $\newline$ Number of arrangements = $\frac{9!}{1! \times 2! \times 2! \times 2! \times 1! \times 1!}$
Calculate Total Factorial: Calculate the factorial of the total number of letters, which is $9!$ . $\newline$ $9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880$
Calculate Letter Factorials: Calculate the factorial for each of the letter frequencies. $\newline$ Since the frequencies are $1$ or $2$ , we only need to calculate $2!$ . $\newline$ $2! = 2 \times 1 = 2$ $\newline$ $1! = 1$ (but we don't really need to calculate this since the factorial of $1$ is $1$ )
Divide Factorials for Arrangements: Divide the total factorial by the product of the factorials of the letter frequencies. $\newline$ Number of arrangements = $\frac{362,880}{1 \times 2 \times 2 \times 2 \times 1 \times 1}$ $\newline$ Number of arrangements = $\frac{362,880}{8}$ $\newline$ Number of arrangements = $45,360$

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