How many chords can be drawn through $21$ points on a circle?

The number of shortest paths from point $A$ to $D$ (as shown in the figure) is:

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 $n$?

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

Suppose $BC$ is a given line segment in the plane and $T$ is a scalene triangle. The number of points $A$ in the plane such that the triangle with vertices $A, B, C$ (in that order) is similar to triangle $T$ is

Out of $10$ points in a plane,$6$ are in a straight line. The number of triangles formed by joining these points is: