Question

The straight lines ${ı}_1,{ı}_{2}and{ı}_3$ are parallel and lie in the same plane. A total of m points are taken on the line ${ı}_1,$ n points on ${ı}_2$, and k points on ${ı}_3$. How many triangles are there whose vertices are at these points?

Answer 1

There are total $n+m+k$ points.

And a triangle can't be formed if all the three points are on the same line.

Total number of ways to select three points ${=}^{n+m+k}{C}_3$

And, number of ways to select three points on the same line ${=}^m{C}_3{+}^n{C}_3{+}^k{C}_3$

Hence, required number of triangles formed

${=}^{n+m+k}{C}_3-{(}^n{C}_3{+}^m{C}_3{+}^k{C}_3)$