There are $n$ distinct points on a circle. If the number of pentagons that can be formed using these points as vertices is equal to the number of triangles that can be formed,then the value of $n$ is:

On the sides $AB, BC, CA$ of a $\Delta ABC$,$3, 4, 5$ distinct points (excluding vertices $A, B, C$) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices is:

The sum of all absolute values of the differences of the numbers $1, 2, 3, \ldots, n$,taken two at a time,i.e.,$\sum \limits_{1 \leq j < i \leq n} |i-j|$ equals:

If $t_n$ denotes the number of triangles formed with $n$ points in a plane,no three of which are collinear,and if $t_{n+1}-t_n=36$,then $n$ is equal to

There are $n$ points in a plane,of which $p$ points are collinear. How many lines can be formed from these points?

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is