In how many ways can $12$ gentlemen sit around a round table so that three specified gentlemen are always together?

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

Six persons $A, B, C, D, E$ and $F$ are to be seated at a circular table facing towards the centre. Find the number of ways this can be done if $A$ must have either $E$ or $F$ on his immediate right and $E$ must have either $F$ or $D$ on his immediate right.

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

$12$ persons are to be arranged at a round table. If two particular persons among them are not to be side by side,the total number of arrangements is

$A$ family consisting of a mother,father,and their $8$ children ($4$ boys and $4$ girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?