This GMAT Math practice question is a problem solving question in counting methods. A very interesting question.

#### Question: 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

- 5
- 21
- 33
- 60
- 6

#### Video Explanation

Scroll for explanatory answer text#### Explanatory Answer

##### List down possibilities: From only 1 box all the way to all 6

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.

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