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Choose the correct option:

Boxes numbered $1, 2, 3, 4$ and $5$ are kept in a row, and they are to be filled with either a red or a blue ball, such that no two adjacent boxes can be filled with blue balls. How many different arrangements are possible, given that all balls of a given colour are exactly identical in all respects?

  1. $8$ 
  2. $10$ 
  3. $15$ 
  4. $22$
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1 Answer

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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, we determine the number of ways of filling the boxes such that the adjacent boxes are filled with blue.

 Two adjacent boxes with blue can be obtained in 4 ways, i.e., (12), (23), (34) and (45).

 Three adjacent boxes  with blue can be obtained in 3 ways, i.e., (123), (234) and (345).

 Four adjacent boxes  with blue can be obtained in 2 ways, i.e., (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

The correct option is D.

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