Question

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 .
(JEE Adv. 2019​

Answer 1

Given: 5 persons - A B C D and E

3 colours - Red, Blue and Green

To Find: Number of ways such that the persons seated in  adjacent seats gets  a different coloured hat

Solution:

Maximum number of hats used of same colour = 2.

Also hats of different colours cannot be used as combination - 1 + 1 + 3 because any three hats cannot be of same colour.  

Therefore, the only combination left is 2 + 2 + 1.

Thus, the three cases of selecting hats are =  2R + 2B + 1G or 2G + 2R + 1B or 2B + 2G + 1R

The person will be selected in ways = 5C1

For remaining 4 people it will be = 3 × 5C1 × 2

= 30

Answer: There can be 30 ways