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

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

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Find the magnitude of a three-dimensional vector

Full solution

Q. What is the total number of different

10

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

\newline

Answer:

Word Frequency: The word STATISTICS has $10$ letters in total, with the following frequency of each letter: $S$ occurs $3$ times, $T$ occurs $3$ times, $A$ occurs $1$ time, $I$ occurs $2$ times, and $C$ occurs $1$ time.
Permutations Formula: To find the total number of different arrangements, we use the formula for permutations of a multiset: $\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}$ , where $n$ is the total number of items to arrange, and $n_1$ , $n_2$ , $\ldots$ , $n_k$ are the frequencies of each distinct item.
Calculation of Frequencies: In this case, $n = 10$ (total letters), $n_1 = 3$ (for S), $n_2 = 3$ (for T), $n_3 = 1$ (for A), $n_4 = 2$ (for I), and $n_5 = 1$ (for C).
Factorial Calculation: Now we calculate the factorial of each of these numbers: $10! = 3,628,800$ , $3! = 6$ , $2! = 2$ , and $1! = 1$ .
Application of Formula: We then apply the formula: $10! / (3! * 3! * 1! * 2! * 1!) = 3,628,800 / (6 * 6 * 1 * 2 * 1) = 3,628,800 / 72 = 50,400$ .
Total Arrangements: Therefore, the total number of different $10$ -letter arrangements that can be formed using the letters in the word STATISTICS is $50,400$ .

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