$A$ regular polygon has $20$ sides. The number of triangles that can be drawn by using the vertices but not using the sides are

$15$ girls are seated at a round table. The number of ways of selecting three girls such that all the three are not seated together is

$A$ polygon has $44$ diagonals. Then the number of sides of the polygon 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 $(T_n)$ that can be formed using the vertices of a polygon with $n$ sides is given by $T_n = \binom{n}{3}$. If $T_{n+1} - T_n = 21$,then what is the value of $n$?

There are $6$ points on a circle. Two triangles are drawn such that they have no common vertices. What is the probability that none of the sides of the triangles intersect?