Five persons \(A, B, C, D\) and \(E\) are seated in a circular arangement, 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 person seated in adjacent seats get different coloured hats, is

Given, 5 persons having 5 hats of colour red, blue and green
\(\therefore \) i.e., 3 colours
Maximum 2 hats of same colour can be used.
\(\therefore \) Number of ways of selecting single colour hat out of 3 colours \(={ }^3 \mathrm{C}_1\) ways.
\(\therefore \) Single colour hat is distributed in 5 persons in \({ }^5 \mathrm{C}_1\) ways
Also, number of ways to distribute alternative coloured hat to adjacent person \(={ }^2 \mathrm{C}_1\)
\(\begin{aligned}
\therefore \text {Required number of ways } & ={ }^3 \mathrm{C}_1 \times{ }^5 \mathrm{C}_1 \times{ }^2 \mathrm{C}_1 \\
& =3 \times 5 \times 2 \\
& =30
\end{aligned}\)