Question

There are three coplanar parallel lines.If any $p$ points are taken on each of the lines, and triangles are fromed by these points then

• A. Number of triangles possible $=3p^2(p-1)$ if one side of triangle lies on a line
• B. Number of triangles possible $=p^2(4p-3)$ if one side of triangle lies on a line
• C. Total number of triangles that can be formed $=p^2(4p-3)$
• D. Total number of triangles that can be formed $=p^3$

Answer 1

Answer: (A), (C)

Total number of triangles that can be formed $=p^2(4p-3)$

The number of triangles with vertices on different lines
$={}^p{C}_1\times {}^p{C}_1\times {}^p{C}_1=p^3$
The number of triangles with two vertices on one line and third vertex on any of the other two lines $={}^3{C}_1\times ({}^p{C}_2\times {}^{2p}{C}_1)$
$=6p\cdot \frac{p(p-1)}{2}$
So, the required number of triangles $=p^3+3p^2(p-1)$
$=p^2(4p-3)$