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 number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is

Answer 1

5 persons having 5 hats of colour red,green,blue i.e 3 colours. Maximum number of same colour hats used $=$ 2, because if we give 3 same colour hats then atleast 2 person having same colour of hat will seat in adjacent seats.
Now, suppose hats used are $B,G,G,R,R.$
So for selecting single colour hat${=}^3{C}_1$ ways,
Then distribute that single colour hat in${=}^5{C}_1$ ways,
Also to distribute alternative coloured hat to adjacent person ${=}^2{C}_1$
$\therefore$ required number of ways ${=}^3{C}_1{\times }^5{C}_1{\times }^2{C}_1=3\times 5\times 2=30.$