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 ensure that no box remains empty, we use the Stirling numbers of the second kind to partition the five balls into three groups (boxes).

2. The number of such partitions is given by \(S(5, 3)\), where \(S(n, k)\) represents the Stirling number of the second kind. Using the formula:

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

3. Since the boxes are of different sizes, we can assign these groups to boxes in \(3! = 6\) ways.

4. Finally, the total number of arrangements is:

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