$p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

Let $ABC$ be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side $AB$, five points $p_5, p_6, p_7, p_8, p_9$ on the side $BC$, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side $AC$. None of these points is a vertex of the triangle $ABC$. The total number of pentagons that can be formed by taking all the vertices from the points $p_1, p_2, \ldots, p_{13}$ is . . . . . . .

If a polygon of $n$ sides has $560$ diagonals,then $n=$

Let $n \geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points blue and the rest red. If the number of red and blue line segments are equal,then the value of $n$ is

If a polygon of $n$ sides has $275$ diagonals,then $n$ is

The number of parallelograms that can be formed from a set of $4$ parallel lines intersecting another set of $3$ parallel lines is: