The number of circular permutations of $n$ distinct objects is......

$A$ family consisting of a mother,father,and their $8$ children ($4$ boys and $4$ girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?

The number of different ways of preparing a garland using $6$ distinct white roses and $5$ distinct red roses such that no two red roses come together is

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?

If $A$ and $B$ are among $20$ persons who sit at random along a round table,then the probability that there are exactly six persons between $A$ and $B$ 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