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

Given that no two persons sitting adjacent, have the hat of same colour, so hats of all colour must be used, also $3$ hats of same colour cannot be used as at least two hats of same colour will become consecutive, so the only combination will be $2, 2, 1.$
Three cases are possible
$\left.\begin{array}{c}a) 2R, 2B, 1G\\ b) 2B, 2G, 1R\\ c) 2R, 2G, 1B\end{array}\right\}-3$ way
Select are person for single colour hat $={}_{ }{}^5{C}_1$ (let say green)

Now remaining $4$ hats can be distributed alternatively in two ways $\left(BRBR \mathrm{or} RBRB\right)$
Total number of ways $=3\times {}_{ }{}^5{C}_1\times 2$
$=30$