If $3$ squares are chosen at random from the $64$ squares of a chess board,then the probability that all of them lie along the same diagonal line is

Statement-$1$: If a polygon has $45$ diagonals,then the number of sides is $10$. Statement-$2$: Out of $n$ non-collinear points,$2$ points can be selected in $^nC_2$ ways.

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 . . . . . . .

The number of diagonals in a polygon is $20$. The number of sides of the polygon is:

How many triangles can be drawn by means of $9$ non-collinear points?

On an $n \times n$ chessboard,the total number of rectangles which are not squares is $350$. Then,the number of white squares on the chessboard is .......