The number of ways in which $5$ beads of different colours form a necklace is

The number of ways of arranging $8$ men and $4$ women around a circular table such that no two women can 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

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:

The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by

$20$ 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?