The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$,all the lines $L_{2n-1}$ are parallel to each other,and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:

There are $n$ points in a plane,of which $p$ points are collinear. How many lines can be formed from these points?

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 diagonals of a polygon is $35$. If $A$ and $B$ are two distinct vertices of this polygon,then the number of all those triangles formed by joining three vertices of the polygon having $AB$ as one of its sides is:

Consider a rectangle $ABCD$ having $5, 6, 7, 9$ points in the interior of the line segments $AB, BC, CD, DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta-\alpha)$ is equal to:

In how many ways can $21$ English books and $19$ Hindi books be placed in a row so that no two Hindi books are together?