Let $X = \{(a, b) \mid a, b \in \mathbb{Z}, 0 \leq a \leq 20, 0 \leq b \leq 15\}$. Then the number of rectangles with vertices in set $X$ is

Ten points lie in a plane such that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions,each containing four of the remaining points,is

$A$ polygon has $35$ diagonals,then the number of its sides is

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n+1} - T_n = 21$,then $n$ equals

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

If $x$ is the number of ways in which six women and six men can be arranged to sit in a row such that no two women are together and if $y$ is the number of ways they are seated around a table in the same manner,then $x: y=$ (in $: 1$)