In how many ways can $12$ gentlemen sit around a round table so that three specified gentlemen are always together?

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

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

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

The number of ways in which $6$ men and $4$ women can be seated around a table so that a particular man and a particular woman never sit adjacent to each other 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?