$4$ gentlemen and $4$ ladies are to be seated at a round table. What is the probability that they sit alternately?

$20$ persons are invited for a party. In how many different ways can they and the host be seated at a circular table,if two particular persons are to be seated on either side of the host?

The number of different ways of preparing a garland using $6$ distinct white roses and $5$ distinct red roses such that no two red roses come together is

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

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

The number of ways,in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together,is