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 $16$ points in a plane out of which $6$ are collinear. How many lines can be drawn by joining these points?

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

The number of diagonals in a convex polygon with $n$ sides is .....

Some couples participated in a mixed doubles badminton tournament. If the number of matches played,such that no couple played in a match,is $840$,then the total number of persons who participated in the tournament is $........$.

If three distinct vertices are chosen at random from the vertices of a cube,then the probability that they form the vertices of an equilateral triangle is: