The number of ways in which $8$ different pearls can be arranged to form a necklace is

Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a cap of one of the three colours red,blue and green,then the number of ways of distributing the caps such that the persons seated in adjacent seats get different coloured caps,is

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

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?

The number of ways in which $6$ men and $5$ women can sit at a round table,if no two women sit together,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?