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

$3$ boys and $3$ girls are to be seated around a circular table. Among the boys,$X$ does not want any girl as a neighbour,and among the girls,$Y$ does not want any boy as a neighbour. How many such arrangements are possible?

If $\alpha = {^m}{C_2}$,then ${^\alpha}{C_2}$ is equal to

Let $S$ be the set of all triangles in the $xy$-plane,each having one vertex at the origin and the other two vertices lying on the coordinate axes with integral coordinates. If each triangle in $S$ has an area of $50$ sq. units,then the number of elements in the set $S$ is:

$8$-digit numbers are formed using the digits $1, 1, 2, 2, 2, 3, 4, 4$. The number of such numbers in which the odd digits do not occupy odd places is:

The number of rectangles that can be obtained by joining four of $12$ vertices of a $12$-sided regular polygon is: