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

Answer 1

Hint: Find the number of ways in which out of 3 colours, the hats with 2 same colours. Then, assume a case where a single coloured hat goes to a person and determine the ways in the other two coloured hats can be distributed. Also, find the ways in which that single coloured hat can go to 5 persons.

Complete step-by-step answer:
Since, persons A, B, C, D and E are seated in a circular arrangement, and hats have to be distributed such that adjacent seats get different coloured hats, then there can be maximum two hats of the same colour.
Suppose we have 5 hats such that there are 2 red, 2 blue and 1 green hat. And these can be selected in 3 ways.
Such as, 2 blue, 2 green and 1 red.
Also, there could be 2 red, 2 blue and 1 green.
But, let us now suppose there are 2 red, 2 blue and 1 green hat .
Now, let us assume that person A gets a green hat, and let B and D both get blue.
Similarly, C and E will get the same colour, say red otherwise the adjacent colour will be the same.
So, there are only two ways possible once a single coloured hat goes to a person.
Now, there are 5 people. Hence, a single coloured hat can go to anyone. Hence, there are 5 such cases.
Now, in order to find all the possible cases, we will multiply all the number of ways of cases discussed above.
$3 \times 2 \times 5 = 30$
Hence, there are a total 30 ways to distribute such hats.

Note: One has to take all the possible cases for solving this question correctly. We multiplied at the end because all cases are necessary conditions. We add when the cases are optional. We need to assume cases such that it satisfies the given conditions.