In how many ways can $10$ people be seated around a circular table such that no two arrangements have the same neighbors? (Clockwise and counter-clockwise arrangements are considered the same.)

$A$ 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?

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.

Find the number of ways in which two Americans,two British,one Chinese,one Dutch,and one Egyptian can sit at a round table so that persons of the same nationality are separated.

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

$A$ couple can sit with $6$ guests around a circular table. In how many ways can they sit if the couple sits in consecutive seats?