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?
10. Combinatorics & Probability
Combinatorics
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Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
How many ways are there to arrange the letters in the word CALCULUS?
A
40,320
B
5040
C
720
D
6
Verified step by step guidance1
First, identify the total number of letters in the word 'CALCULUS'. There are 8 letters.
Next, determine if there are any repeated letters. In 'CALCULUS', the letter 'C' appears twice, and the letter 'L' appears twice.
To find the number of distinct arrangements, use the formula for permutations of 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 all possible arrangements of 8 letters, and \( 2! \) for each of the repeated letters 'C' and 'L'.
Calculate the factorials: \( 8! = 40320 \), \( 2! = 2 \). Then, divide: \( \frac{40320}{2 \times 2} \) to find the number of distinct arrangements.
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