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## 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 (GMAT-MATHS)

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 (GMAT-MATHS)

(A) 5 (B) 21
(C) 33 (D) 60
(E) 40

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If only one of the boxes has a green ball, it can be any of the 6 boxes. So, this can be
achieved in 6 ways.
If two of the boxes have green balls and then there are 5 consecutive sets of 2 boxes. 12,
23, 34, 45, 56.
Similarly, if 3 of the boxes have green balls, there will be 4 options.
If 4 boxes have green balls, there will be 3 options.
If 5 boxes have green balls, then there will be 2 options.
If all 6 boxes have green balls, then there will be just 1 option.
Total number of options = 6 + 5 + 4 + 3 + 2 + 1 = 21

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Here option B is correct.

Explanation:

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.

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