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?

• A. 120
• B. 11,880
• C. 27,720
• D. 479,001,600

Answer: (C)

Verified step by step guidance

1. First, recognize that the problem involves arranging books where some are indistinguishable from others due to their color. This is a permutation problem with repetition.
2. The formula for permutations of a multiset 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 indistinguishable group.
3. In this problem, you have a total of 12 books: 4 blue, 3 yellow, and 5 white. So, \( n = 12 \), \( n_1 = 4 \), \( n_2 = 3 \), and \( n_3 = 5 \).
4. Substitute these values into the formula: \( \frac{12!}{4! \times 3! \times 5!} \).
5. Calculate the factorials: \( 12! \) is the factorial of 12, \( 4! \) is the factorial of 4, \( 3! \) is the factorial of 3, and \( 5! \) is the factorial of 5. Simplify the expression to find the number of different ways to arrange the books.