Question

Five balls of different colors are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is:

• A. 160
• B. 140
• C. 180
• D. 150

Hint:

For problems involving partitions and grouping with restrictions, use Stirling numbers and factorials for proper counting

Answer 1

The Correct Option is D

1. To avoid empty boxes, we apply Stirling numbers of the second kind to divide the five balls into three groups (boxes).

2. The count of these partitions is \(S(5, 3)\), where \(S(n, k)\) denotes the Stirling number of the second kind. Using the formula:

\(S(5, 3) = 25.\)

3. As the boxes vary in size, we can arrange the groups into boxes in \(3! = 6\) ways.

4. Therefore, the total number of arrangements is:

\(S(5, 3) \cdot 3! = 25 \cdot 6 = 150.\)