Boxes numbered 1,2,3,4 and 5 are kept in a row, and they are necessarily 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 balls of a given colour are exactly identical in all respects?
Boxes numbered 1,2,3,4 and 5 are kept in a row, and they are necessarily 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 balls of a given colour are exactly identical in all respects?
The correct option is D 22
Total number of box=5
The number of ways of filling 5boxes
with red or blue balls=25=32
If we have 2 adjacent boxes with blue, it can be
done in 4 ways=(12)(23)(34) and (45).
If
we have 3 adjacent boxes with blue, it can be done in 3 ways=(123)(234) and (345).
If
we have 4 adjacent boxes with blue, it can be done in 2 ways=(1234) and (2345).
All
5 boxes can have blue in only one way.
Hence,
the no of ways of filling up the boxes such that no two adjacent boxes have
blue =32− (total number of ways of filling the boxes such that adjacent boxes
have blue)
=32−(10)
=32−10
=22
Total number of box=5
The number of ways of filling 5boxes with red or blue balls=25=32
If we have 2 adjacent boxes with blue, it can be done in 4 ways=(12)(23)(34) and (45).
If we have 3 adjacent boxes with blue, it can be done in 3 ways=(123)(234) and (345).
If we have 4 adjacent boxes with blue, it can be done in 2 ways=(1234) and (2345).
All 5 boxes can have blue in only one way.
Hence, the no of ways of filling up the boxes such that no two adjacent boxes have blue =32− (total number of ways of filling the boxes such that adjacent boxes have blue)
=32−(10)
=32−10
=22
