Question 73

They are six boxes are numbered 1, 2, 3, 4, 5, 6. Each box is to be filled with either red or green balls in such a way that at least one box containing green balls are consecutively numbered .The total number of days in which these can be done is

Solution

Case 1: Total ways when one box contains one greenÂ ball = {1}. {2}, {3}, {4}, {5}, {6} =Â 6 ways

Case 2: Total ways when two boxes contain oneÂ green ball = {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6} = 5 ways

Case 3: Total ways when three boxes contain one green ball = {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6} = 4 ways

Case 4: Total ways when four boxes contain one green ball = {1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6} =Â 3 ways

Case 5: Total ways when five boxes contain one green ball = {1, 2, 3, 4, 5}, {2, 3, 4, 5, 6} =Â 2 ways

Case 5: Total ways when all the sixÂ boxes contain one green ball = {1, 2, 3, 4, 5, 6} = 1 way

Tota; ways = 6 + 5 + 4 + 3 +Â 2 +Â 1 = 21

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