The number of diagonals in a polygon of $m$ sides is

Let $t_n$ denote the number of integral-sided triangles with distinct sides chosen from $\{1, 2, 3, \ldots, n\}$. Then,$t_{20} - t_{19}$ equals

The number of triangles whose vertices are the points $(x, y)$ in the $XY$-plane with integer coordinates satisfying $0 \leq x \leq 4$ and $0 \leq y \leq 4$ is:

Out of $10$ points in a plane,$6$ are collinear. How many triangles can be formed by joining these points?

In a regular $15$-sided polygon with all its diagonals drawn,a diagonal is chosen at random. The probability that it is neither a shortest diagonal nor a longest diagonal is

Three out of $6$ vertices of a regular hexagon are chosen at random. The probability that the triangle formed with these three vertices is an equilateral triangle is