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?
21. Combinatorics and Probability
Combinatorics
Multiple Choice
How many ways are there to arrange the letters in the word CALCULUS?
A
40,320
B
5,040
C
720
D
6
Verified step by step guidance1
Identify the total number of letters in the word 'CALCULUS'. There are 8 letters.
Determine if there are any repeated letters. 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: \( \frac{8!}{2! \times 2!} \). Calculate \( 8! \) for the total permutations of the letters, and divide by the factorials of the repeated letters.
Simplify the expression to find the number of unique arrangements of the letters in 'CALCULUS'.
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