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?
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Multiple Choice
You want to arrange the books on your bookshelf by color. How many different ways could you arrange 12 books if 4 of them have a blue cover, 3 are yellow, and 5 are white?
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Verified step by step guidance1
First, recognize that this is a permutation problem with groups of identical items. You have 12 books in total, with 4 blue, 3 yellow, and 5 white.
The formula for permutations of items with identical groups is given by: \( \frac{n!}{n_1! \times n_2! \times n_3!} \), where \( n \) is the total number of items, and \( n_1, n_2, n_3 \) are the counts of each identical group.
Substitute the values into the formula: \( n = 12 \), \( n_1 = 4 \) (blue books), \( n_2 = 3 \) (yellow books), \( n_3 = 5 \) (white books).
Calculate the factorials: \( 12! \), \( 4! \), \( 3! \), and \( 5! \).
Divide \( 12! \) by the product of \( 4! \times 3! \times 5! \) to find the number of different ways to arrange the books.
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