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

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

In how many ways can $10$ people be seated around a circular table such that no two arrangements have the same neighbors? (Clockwise and counter-clockwise arrangements are considered the same.)

$12$ persons are to be arranged at a round table. If two particular persons among them are not to be side by side,the total number of arrangements is

$8$ teachers and $4$ students are sitting around a circular table at random. Then,the probability that no two students sit together is:

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: