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What is the total number of different 13 -letter arrangements that can be formed using the letters in the word MATHEMATICIAN?
Answer:

What is the total number of different $13$ -letter arrangements that can be formed using the letters in the word MATHEMATICIAN? $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. What is the total number of different

13

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

\newline

Answer:

Frequency of Each Letter: Determine the frequency of each letter in the word MATHEMATICIAN. $\newline$ $M$ appears $2$ times. $\newline$ $A$ appears $3$ times. $\newline$ $T$ appears $2$ times. $\newline$ $H$ appears $1$ time. $\newline$ $E$ appears $1$ time. $\newline$ $2$ $0$ appears $2$ times. $\newline$ $2$ $2$ appears $1$ time. $\newline$ $2$ $4$ appears $1$ time.
Calculate Total Factorial: Calculate the factorial of the total number of letters, which is $13$ . $\newline$ $13! = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Factorial of Frequency: Calculate the factorial of the frequency of each letter to account for indistinguishable arrangements. $\newline$ M: $2! = 2 \times 1$ $\newline$ A: $3! = 3 \times 2 \times 1$ $\newline$ T: $2! = 2 \times 1$ $\newline$ H: $1! = 1$ $\newline$ E: $1! = 1$ $\newline$ I: $2! = 2 \times 1$ $\newline$ C: $1! = 1$ $\newline$ N: $1! = 1$
Permutations Formula: Use the formula for permutations of a multiset to find the total number of different arrangements. $\newline$ The formula is: $\newline$ Total arrangements = $\frac{13!}{(2! \times 3! \times 2! \times 1! \times 1! \times 2! \times 1! \times 1!)}$
Perform Calculations: Perform the calculations using the values from the previous steps. $\newline$ Total arrangements = $\frac{13!}{2! \times 3! \times 2! \times 1! \times 1! \times 2! \times 1! \times 1!}$ $\newline$ Total arrangements = $\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{((2 \times 1) \times (3 \times 2 \times 1) \times (2 \times 1) \times 1 \times 1 \times (2 \times 1) \times 1 \times 1)}$ $\newline$ Total arrangements = $\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(2 \times 6 \times 2 \times 1 \times 1 \times 2 \times 1 \times 1)}$ $\newline$ Total arrangements = $\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(48)}$ $\newline$ Total arrangements = $\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 1)}{(2)}$ $\newline$ Total arrangements = $\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3)}{(2)}$ $\newline$ Total arrangements = $\frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{2}$ $\newline$ Total arrangements = $1037836800$

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