Question

A person has 3 different bags & 4 different books. The number of ways in which he can put these books in the bags so that no bag is empty, is

• A. 36
• B. 24
• C. 32
• D. 30

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To distribute 4 distinct books into 3 distinct bags such that no bag is empty, we must partition the 4 books into 3 non-empty groups and then assign those groups to the bags.

Step 2: Key Formula or Approach:
The only possible distribution of 4 books into 3 bags where each bag gets at least one book is: **(2, 1, 1)**.

Step 3: Detailed Explanation:
1. **Grouping the books:** Number of ways to split 4 distinct books into groups of size 2, 1, and 1: \[ \text{Ways} = \frac{4!}{2!1!1!} \times \frac{1}{2!} = \frac{24}{2 \times 2} = 6 \] *(Note: We divide by 2! because there are two groups of the same size, 1).* 2. **Assigning groups to bags:** Since the 3 bags are distinct, we multiply by the number of arrangements of the 3 groups: \[ \text{Total ways} = 6 \times 3! = 6 \times 6 = 36 \]

Step 4: Final Answer:
The total number of ways is 36.