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

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

\newline

Answer:

Count Letters: First, count the number of each letter in the word PRECIPITOUS. $P$ appears $2$ times, $R$ appears $1$ time, $E$ appears $1$ time, $C$ appears $1$ time, $I$ appears $2$ times, $2$ $0$ appears $1$ time, $2$ $2$ appears $1$ time, $2$ $4$ appears $1$ time, $2$ $6$ appears $1$ time.
Formula for Arrangements: The formula for the number of arrangements of a word with repeated letters is $n! / (p_1! * p_2! * ... * p_k!)$ , where $n$ is the total number of letters, and $p_1, p_2, ..., p_k$ are the frequencies of the repeated letters.
Calculate Total Factorial: Calculate the factorial of the total number of letters, which is $11!$ for PRECIPITOUS. $\newline$ $11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Calculate Factorial for Repeated Letters: Calculate the factorial for the repeated letters $P$ and $I$ , which appear $2$ times each. $\newline$ $2! = 2 \times 1$
Divide Factorials: Now, divide $11!$ by the product of the factorials of the frequencies of the repeated letters. $\newline$ So, the number of different arrangements is $\frac{11!}{(2! \cdot 2!)}$ .
Perform Calculation: Perform the calculation: $11! / (2! \times 2!) = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1}$
Simplify Calculation: Simplify the calculation: $\frac{11!}{(2! \times 2!)} = \frac{(11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3)}{(2 \times 2)}$
Finish Calculation: Finish the calculation: $11! / (2! \times 2!) = (11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3) / 4$
Final Answer: The final answer is $11! / (2! \times 2!) = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 / 4 = 9979200$ .

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