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

The number of ways that $8$ beads of different colours can be strung as a necklace is

The number of ways in which $5$ male and $2$ female members of a committee can be seated around a round table so that the two female members are not seated together is:

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?

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

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.