In how many ways can $15$ members of a council sit along a circular table,when the Secretary is to sit on one side of the Chairman and the Deputy Secretary on the other side?

$4$ gentlemen and $4$ ladies are to be seated at a round table. What is the probability that they sit alternately?

In how many ways can $7$ women and $7$ men be seated around a circular table such that each woman is seated next to a man?

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.)

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:

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