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. Identify the total number of books to arrange, which is 12.
2. Recognize that the books are grouped by color: 4 blue, 3 yellow, and 5 white.
3. Use the formula for permutations of multiset: \( \frac{n!}{n_1! \times n_2! \times n_3!} \), where \( n \) is the total number of items, and \( n_1, n_2, n_3 \) are the counts of each distinct item.
4. Substitute the values into the formula: \( \frac{12!}{4! \times 3! \times 5!} \).
5. Calculate the factorials: \( 12! \), \( 4! \), \( 3! \), and \( 5! \), and then divide as per the formula to find the number of arrangements.