$n$ gentlemen can be made to sit on a round table in how many ways?

If $20$ beads of two different colors are to be arranged in a necklace such that they alternate,and there are $10$ beads of each color,then what is the number of ways to arrange them?

If $3$ sisters and $8$ brothers are together playing a game,then the number of ways in which all the sisters and brothers are to be seated around a circle such that all the $3$ sisters are not seated together 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:

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?

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