The number of ways in which $3$ boys and $4$ girls can be seated around a circular table if no two boys sit together is

In how many ways can $10$ people be seated around a circular table such that no two arrangements have the same neighbors? (Clockwise and counter-clockwise arrangements are considered the same.)

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

The number of ways in which $6$ men and $5$ women can sit at a round table,if no two women sit together,is:

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

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