$20$ 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?

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:

The number of ways in which $5$ boys and $4$ girls can sit around a circular table such that no two girls sit together is:

The number of ways,in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together,is

$4$ 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 $5$ men and $2$ women be seated around a circular table such that the two women do not sit together?