Multiple Choice
A student formed a club at their school. They have 13 members, and need to elect a president, vice president, and treasurer. How many ways are there to fill these officer positions?
4. Probability
Counting
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How many ways are there to arrange the letters in the word CALCULUS?
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Verified step by step guidance1
Identify the total number of letters in the word 'CALCULUS'. There are 8 letters in total.
Determine if there are any repeated letters in the word. In 'CALCULUS', the letter 'C' appears twice, and the letter 'L' appears twice.
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 letters, and \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.
Substitute the values into the formula: \( \frac{8!}{2! \times 2!} \). Here, \( 8! \) accounts for the total number of letters, and \( 2! \) for each of the repeated letters 'C' and 'L'.
Calculate the factorials: \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) and \( 2! = 2 \times 1 \). Then, divide the result of \( 8! \) by the product of the factorials of the repeated letters to find the number of unique arrangements.
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