Five persons A, B, C, D and E are seated in a circular arrangement. If

1.	Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is _____
Answer» Given that no two persons sitting adjacent, have hats of same colour. So hats of all colours must be used. Also hats of different colours cannot be used in 1 + 1 + 3 combination because any three hats cannot be of same colour. Therefore only combination left is 2 + 2 + 1. There are a total of 3 cases of selecting hats which are 2R + 2B + 1G or 2B + 2G + 1R or 2G + 2R + 1B To distribute these hats first we select a person in ⁵C₁ ways and distribute that hat which is one of it's colour. Then for remaining four persons there are two ways of distributing hats of alternate colours. So total ways will be equal to 3 × ⁵C₁ × 2 = 30

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is _____

Answer»

Given that no two persons sitting adjacent, have hats of same colour. So hats of all colours must be used. Also hats of different colours cannot be used in 1 + 1 + 3 combination because any three hats cannot be of same colour.

Therefore only combination left is 2 + 2 + 1.

There are a total of 3 cases of selecting hats which are

2R + 2B + 1G or 2B + 2G + 1R or 2G + 2R + 1B

To distribute these hats first we select a person in ⁵C₁ ways and distribute that hat which is one of it's colour. Then for remaining four persons there are two ways of distributing hats of alternate colours. So total ways will be equal to 3 × ⁵C₁ × 2 = 30

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