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

In how many ways can $5$ boys and $5$ girls be seated around a circular table such that no two girls are together?

The number of ways that $8$ beads of different colours can be strung as a necklace is

If $11$ members of a committee sit at a round table so that the President and Secretary always sit together,then the number of arrangements is

Find the number of ways in which two Americans,two British,one Chinese,one Dutch,and one Egyptian can sit at a round table so that persons of the same nationality are separated.

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