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Difficulty: MediumBasic Probability and Counting Methods

A school library receives a shipment of 1212 new books, consisting of 55 science fiction novels, 44 historical fiction novels, and 33 biography books. A librarian wants to choose 33 books to feature on a display shelf. How many different combinations of 33 books can be formed that contain exactly 22 science fiction novels and 11 biography book?

Answer: 30 combinations

Answer

The total number of combinations containing exactly 2 science fiction novels and 1 biography book is 30.
To find the total number of combinations, first determine the number of ways to choose 22 science fiction novels from 55 available: (52)=5×42×1=10\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10. Next, determine the number of ways to choose 11 biography book from 33 available: (31)=3\binom{3}{1} = 3. Finally, apply the Fundamental Counting Principle by multiplying the counts together: 10×3=3010 \times 3 = 30.

Step-by-Step Solution

1
Calculate the combinations for selecting the science fiction novels.
(52)=5×42×1=10\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10
Select 22 novels out of the 55 available science fiction novels without regard to order.
2
Calculate the combinations for selecting the biography book.
(31)=3\binom{3}{1} = 3
Select 11 book out of the 33 available biography books.
3
Apply the Fundamental Counting Principle to find total combinations.
10×3=3010 \times 3 = 30
Multiply the combinations calculated for each independent selection.

Key Concept

Combinations and the Fundamental Counting Principle
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Topics

Basic Probability and Counting Methods

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