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What is the total number of different 11-letter arrangements that can be formed using the letters in the word RESPIRATION?
Answer:

What is the total number of different $11$ -letter arrangements that can be formed using the letters in the word RESPIRATION? $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. What is the total number of different

11

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

\newline

Answer:

Count Letters Frequency: Determine the total number of letters and the frequency of each letter in the word RESPIRATION. $\newline$ The word RESPIRATION has $11$ letters in total. The frequency of each letter is as follows: $\newline$ R - $2$ times $\newline$ E - $1$ time $\newline$ S - $1$ time $\newline$ P - $1$ time $\newline$ I - $2$ times $\newline$ A - $1$ time $\newline$ T - $1$ time $\newline$ O - $1$ time $\newline$ N - $1$ time
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 RESPIRATION, this becomes: $\newline$ Number of arrangements = $\frac{11!}{2! \times 1! \times 1! \times 1! \times 2! \times 1! \times 1! \times 1! \times 1!}$
Calculate Total Factorial: Calculate the factorial of the total number of letters, which is $11!$ . $\newline$ $11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Calculate Letter Factorials: Calculate the factorial of the frequency of each letter. $\newline$ $2! = 2 \times 1$ $\newline$ $1! = 1$ (for all the letters that appear only once)
Substitute and Calculate: Substitute the factorial values into the formula and calculate the number of arrangements. $\newline$ Number of arrangements = $11! / (2! * 1! * 1! * 1! * 2! * 1! * 1! * 1! * 1!)$ $\newline$ = $(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * 1 * 1 * 1 * (2 * 1) * 1 * 1 * 1 * 1)$ $\newline$ = $(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 2)$ $\newline$ = $(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / 4$ $\newline$ = $11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 / 4$ $\newline$ = $11 * 10 * 9 * 8 * 7 * 6 * 5 * 3$ $\newline$ = $11 * 10 * 9 * 8 * 7 * 6 * 15$ $\newline$ = $11 * 10 * 9 * 8 * 7 * 90$ $\newline$ = $11 * 10 * 9 * 8 * 630$ $\newline$ = $11 * 10 * 9 * 5040$ $\newline$ = $(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * 1 * 1 * 1 * (2 * 1) * 1 * 1 * 1 * 1)$ $0$ $\newline$ = $(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * 1 * 1 * 1 * (2 * 1) * 1 * 1 * 1 * 1)$ $1$ $\newline$ = $\(4989600\)

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