Question

The straight lines $l_2||l_2||l_3$ lies in the same plane. A total of m points are taken on $l_1,n$ points on $l_2$ and k points on $l_3$, then the maximum number of triangles formed with vertices at these points are

• A. ${}^{m+n+k}{C}_3-{(}^m{C}_3{+}^n{C}_3{+}^k{C}_3)$
• B. ${}^m{C}_3{+}^n{C}_3{+}^k{C}_3$
• C. ${}^{m+n+k}{C}_3$
• D. None of these

Answer 1

The correct option is A ${}^{m+n+k}{C}_3-{(}^m{C}_3{+}^n{C}_3{+}^k{C}_3)$

Total Number of points $=m+n+k$
$\therefore$ Number of $\Delta$'s formed $=m+n+k_{c_3}$
Now m,n,k are lies on the lines
$l_1,l_2,l_3$ respectively means these are collinear points for the line $l_1,l_2,l_3$ respectively these point do not form any triangle.
$\therefore$ Number of dummy triangle ${}^m{C}_3{,}^n{C}_3{,}^k{C}_3$
$\therefore$ actual number of triangle ${=}^{m+n+k}{C}_3-{(}^m{C}_3{+}^n{C}_3{+}^k{C}_3)$