Permutation and CombinationMedium
Question
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 balls. How many different arrangements are possible, given that the balls of a given colour are exactly identical in all respects ?
Options
A.8
B.10
C.13
D.22
Solution
Make cases when all 5 boxes are filled by :
Case 1 : identical 5 red balls
5C5 1 way
Case 2 : 4 identical red balls and 1 blue ball
5C1 = 5 ways
Case 3 : 3 blue and 2 red balls i.e. xRxRx
⇒ 4 gaps, for 2 blue balls
∴ 4C2 = 6 ways
Case 4 : 2 red and 3 blue balls i.e. xRxRx
gaps, 3 blue balls
∴ Total number of ways are 1 + 5 + 6 + 1 = 13 ways
Case 1 : identical 5 red balls
5C5 1 way
Case 2 : 4 identical red balls and 1 blue ball
5C1 = 5 ways
Case 3 : 3 blue and 2 red balls i.e. xRxRx
⇒ 4 gaps, for 2 blue balls
∴ 4C2 = 6 ways
Case 4 : 2 red and 3 blue balls i.e. xRxRx
gaps, 3 blue balls
∴ Total number of ways are 1 + 5 + 6 + 1 = 13 ways
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