Question

There are $p,q,r$ points on three parallel lines $L_1,L_{2}and{L}_3$all of which lie in one plane. The number of triangles which can be formed with vertices at these points is

• A. ${}^{p+q+r}{C}_3$
• B. ${}^{p+q+r}{C}_3{–}^p{C}_3{–}^q{C}_3{-}^r{C}_3$
• C. ${}^p{C}_3{+}^q{C}_3{+}^r{C}_3$
• D. none of these

Answer 1

The correct option is B

${}^{p+q+r}{C}_3{–}^p{C}_3{–}^q{C}_3{-}^r{C}_3$

Explanation for the correct option

Finding the number of triangles which can be formed with vertices at these points

Given: There are $p,q,r$ points on three parallel lines $L_1,L_{2}and{L}_3$all of which lie in one plane.

Total number of points are $p+q+r.$

Number of triangles is${}^{p+q+r}{\mathrm{ℂ}}_3$
The points on $L_1$will not form a triangle.

Hence exclude ${}^p{\mathrm{ℂ}}_3$​ and similarly exclude ${}^q{\mathrm{ℂ}}_3$​ and ${}^r{\mathrm{ℂ}}_3$​ for q points on $L_2$ and $r$ points on $L_3$

Hence the correct answer is option (B)