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

The number of ways of arranging $8$ men and $4$ women around a circular table such that no two women can sit together is

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

$A$ person invites $8$ guests to a dinner and places $5$ of them at one table and the remaining $3$ at another,both the tables being round. The number of ways in which the guests can be arranged is

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

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