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

If $A$ and $B$ are among $20$ persons who sit at random along a round table,then the probability that there are exactly six persons between $A$ and $B$ is

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.

$A$ 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

The number of ways in which $6$ boys and $4$ girls can be seated around a round table such that $2$ special boys and a special girl never sit together is