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:

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:

How many triangles can be drawn by means of $9$ non-collinear points?

Let $P_1, P_2, \ldots, P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_i, P_j, P_k$ such that $i+j+k \neq 15$ is

There are $n$ distinct points on the circumference of 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 what is the value of $n$?

Ten points lie in a plane such that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions,each containing four of the remaining points,is