In how many ways can $7$ women and $7$ men be seated around a circular table such that each woman is seated next to a man?

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.

Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a cap of one of the three colours red,blue and green,then the number of ways of distributing the caps such that the persons seated in adjacent seats get different coloured caps,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:

$8$ teachers and $4$ students are sitting around a circular table at random. Then,the probability that no two students sit together is:

In how many ways can $4$ beads be selected and arranged in a necklace from $6$ distinct beads?