In how many ways can $5$ keys be put in a ring?

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

$20$ persons are invited for a party. In how many different ways can they and the host be seated at a circular table,if two particular persons are to be seated on either side of the host?

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

Five boys and five girls are to be seated around a circular table. The number of ways in which they can be seated such that no two boys and no two girls are together is:

Let $m$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a line alternately,and $n$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a circle so that no two boys are together. If $m = kn$,then the value of $k$ is