Boxes numbered 1, 2, 3, 4 and 5 are kept in a row and they which are to be filled with either a red or a blue ball, such that no two adjacent boxes can be filled with blue balls. then, how many different arrangements are possible, given that all balls of a given colour are exactly identical in all respects ?
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Total number of ways of filling the 5 boxes numbered as (1, 2, 3, 4 and 5) with either blue or red balls = 25 = 32.
Now, let us determine the number of ways of filling the boxes such that the adjacent boxes are filled with blue.
If we decide to have 2 adjacent boxes with blue, it can be done in 4 ways, viz. (12), (23), (34) and (45).
If we decide to have 3 adjacent boxes filled with blue, it can be done in 3 ways, viz. (123), (234) and (345).
If we decide to have 4 adjacent boxes filled with blue, it can be done in 2 ways, viz. (1234) and (2345).
And all 5 boxes can have blue in only 1 way.
Hence, the total number of ways of filling the boxes such that adjacent boxes have blue = (4 + 3 + 2 +1) = 10.
Hence, the number of ways of filling up the boxes such that no two adjacent boxes have blue = 32 - 10 = 22
Now, let us determine the number of ways of filling the boxes such that the adjacent boxes are filled with blue.
If we decide to have 2 adjacent boxes with blue, it can be done in 4 ways, viz. (12), (23), (34) and (45).
If we decide to have 3 adjacent boxes filled with blue, it can be done in 3 ways, viz. (123), (234) and (345).
If we decide to have 4 adjacent boxes filled with blue, it can be done in 2 ways, viz. (1234) and (2345).
And all 5 boxes can have blue in only 1 way.
Hence, the total number of ways of filling the boxes such that adjacent boxes have blue = (4 + 3 + 2 +1) = 10.
Hence, the number of ways of filling up the boxes such that no two adjacent boxes have blue = 32 - 10 = 22
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