Question

Five person A, B, C, D and E are seated in a circular arrangement.

Answer 1

Five person$A,B,C,D\&E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colors $red,blue\&green$, then the number of ways of distributing the hats such that the persons

seated in adjacent seats get different coloured hats is,

Maximum number of hats used of same color are $2$.

They cannot be $3$ otherwise at least $2$ hats of same color are consecutive.

Now the hats used are consider as $BBGGR$.

Which can be selected in 3 ways.

It can be $RGGBBorRRGBBorGGRRB$.

The number of ways of distributing blue hat (single one) in $5$ persons equal to $5$.

Now either position $B\&D$ are filled by green hats and $C\&E$ are filled by Red hats or $B\&D$are filled by Red hats and $C\&E$ are filled by Green hats.

So, only $2$ ways are possible.

Hence, number of ways $=3\times 5\times 2=30$ ways.