GMAT Practice Questions | Combination Q4

This GMAT Math practice question is a problem solving question in counting methods. A very interesting GMAT 650 to 700 level sample question.

Question 4: There are 6 boxes numbered 1, 2,....6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is ___

1. 5
2. 21
3. 33
4. 60
5. 6

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Video Explanation

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Explanatory Answer | GMAT Permutation Combination Q4

List down possibilities: From only one box containing a green ball to all six boxes containing green balls.

If only one of the boxes has a green ball, it can be any of the 6 boxes. So, we have 6 possibilities.
If two of the boxes have green balls and then there are 5 consecutive sets of 2 boxes. 12, 23, 34, 45, 56.
If 3 of the boxes have green balls, there are 4 possibilities: 123, 234, 345, 456.
If 4 boxes have green balls, there are 3 possibilities: 1234, 2345, 3456.
If 5 boxes have green balls, there are 2 possibilities: 12345, 23456.
If all 6 boxes have green balls, there is just 1 possibility.

Total number of possibilities = 6 + 5 + 4 + 3 + 2 + 1 = 21.

Choice B is the correct answer.