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Permutation Combination : Practice Question

Question 6

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) 27

Correct Answer is 21 - Choice 2

Explanation

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 arrangement possible. i.e., the two boxes can one of 1-2 or 2-3 or 3-4 or 4-5 or 5-6.

If 3 of the boxes have green balls, there will be 4 options in which the 3 boxes are in consecutive positions. i.e., 1-2-3 or 2-3-4 or 3-4-5 or 4-5-6

If 4 boxes have green balls, there will be 3 options. i.e., 1-2-3-4 or 2-3-4-5 or 3-4-5-6

If 5 boxes have green balls, then there will be 2 options. i.e., 1-2-3-4-5 or 2-3-4-5-6

If all 6 boxes have green balls, then there will be just 1 options.

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

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