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.)

Let $m$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a line alternately,and $n$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a circle so that no two boys are together. If $m = kn$,then the value of $k$ is

If a party of $n$ persons sit at a round table,then the odds against two specified individuals sitting next to each other are

In how many ways can $6$ men and $5$ women be seated around a circular table if no two women are to be seated together?

The number of circular permutations of $n$ different objects 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: