Question: Five balls of different colors are to be placed in three boxes of different sizes. Each box can hold all five. In how many different ways can we place the balls so that no box remains empty?

1. 100
2. 125
3. 150
4. 175
5. 200

Correct Answer: C
Solution and Explanation
Approach Solution 1:

Possible distributions are:
3-1-1:
Choose the balls: 5c3 x 2c1 x 1c1 = 20
Decide the arrangements: 3!/2! = 3
Total = 20*3 = 60
Or
2-2-1
Choose the balls: 5c2 x 3c2 x 1c1 = 30
Decide the arrangements: 3!/2! = 3
Total = 30*3 = 90
Total = 60+90 = 150

Approach Solution 2:

Each ball has 3 options (boxes)
Total ways = 3^5 = 243

Cases to be removed:

1. Cases where all balls are in 1 box = 3
2. Cases where all balls are in 2 boxes:

Choose 2 boxes = 3c2 = 3
Each ball has 2 options = 2^5 = 32
But, of these, there are 2 cases when the balls are in only 1 box; these we should ignore as we considered this in 'Case 1'; ie. 32-2 = 30
So total = 3c2 x (32-2) = 90
Hence, required answer
= 243 - 3 - 90 = 150

Approach Solution 3:
Since each box must have at least one ball, we see that there are only two cases to consider:
Case 1: (3, 1, 1) or (1, 3, 1) or (1, 1, 3)
For (3, 1, 1), we have 5C3 x 2C1 x 1C1 = 10 x 2 x 1 = 20 ways to choose the balls.
For (1, 3, 1), we have 5C1 x 4C3 x 1C1 = 5 x 4 x 1 = 20 ways to choose the balls.
For (1, 1, 3), we have 5C1 x 4C1 x 3C3 = 5 x 4 x 1 = 20 ways to choose the balls.

Case 1 yields a total of 20 + 20 + 20 = 60 ways.
Case 2: (1, 2, 2) or (2, 1, 2,) or (2, 2, 1)
For (1, 2, 2), we have 5C1 x 4C2 x 2C2 = 5 x 6 x 1 = 30 ways to choose the balls.
Note that the two remaining choices (2, 1, 2,) or (2, 2, 1) yield 30 ways and 30 ways, respectively.

Thus, Case 2 yields 30 + 30 + 30 = 90 ways.
The total number of ways to place the balls such that no box remains empty is 60 + 90 = 150.

“Five balls of different colors are to be placed in three boxes of”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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