Question

The straight lines L1, L2, L3 are parallel and lie in the same plane. A total number of $m$ points are taken on L1.$n$ points on L2 and $k$ points on L3. Then maximum number of triangles formed with vertices at these points are:

• A. ${}^{m+n+k}{C}_3$
• B. ${}^{m+n+k}{C}_3-{(}^m{C}_3{+}^n{C}_3{+}^k{C}_3)$
• C. ${(}^m{C}_3{+}^n{C}_2{+}^k{C}_3)$
• D. ${}^m{C}_2{.}^n{C}_2{+}^n{C}_1{.}^k{C}_2{+}^n{C}_2{.}^m{C}_1$

Answer 1

Answer: (B)

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

The straight lines L1,L2,L3 are parallel and lie in the same plane. A total number of m points are taken on L1. n points on L2 and k points on L3.

Then maximum number of triangles formed with vertices at these points are$=$ (Selecting 3 points from the m+n+k points to form a triangle)$-$ [(m points lie on same line so selecting 3 points from this will not form a triangle) $+$(n points lie on same line so selecting 3 points from this will not form a triangle)$+$(k points lie on same line so selecting 3 points from this will not form a triangle)]

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