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 colors red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different colored hats, is
What I try:
Clockwise sense $(A,B,C,D,E)$ in a circle:
If $A$ is given red, then $B$ has two possibilities and $E$ has one possibility.
$\bullet$ If $A$ is $\text{Red}$ and $B$ is $\text{Green}$. Then $C,E$ has only one possibility ($\text{Blue}$) and $D$ has $2$ possibilities.
$\bullet$ If $A$ is $\text{Red}$ and $B$ is $\text{Blue}$. Then $C,E$ has only one possibility ($\text{Green}$) and $D$ has $2$ possibilities.
And we can arrange $A$ as $\text{Red,Green,Blue}(3)$ possibilities.
So we have total $3\cdot 4=12$
But answer is $30$.
How do I solve it? Help me please.
Your starting point is wrong. E is not adjacent to B, only to A, so if you assign colours to A and B, there are still two possibilities for E. But we need to look a little more carefully:
If we give A red and B blue, then there are 5 arrangements: RBRBG, RBRGB, RBGRB, RBGRG, RBGBG. But there are 6 ways of giving different coloured hats to A and B. Each will give 5 different arrangements. So 30 in all.