Question

The straight lines $I_1,I_2,I_3$ are parallel and lie in the same plane. A total number of $m$ points on $I_1$; $n$ points on $I_2$; $k$ points on $I_3$, the 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. ${}^m{C}_3{+}^n{C}_3{+}^k{C}_3$
• D. None of these

Answer 1

Answer: (B)

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

Toatl number of points are $m+n+k$
the ${\Delta }^{\prime }s$ formed by these points ${=}^{m+n+k}{C}_3$
Points on the same line gives no triangle
Such ${\Delta }^{\prime }s$ are ${}^m{C}_3{+}^n{C}_3{+}^k{C}_3$
Therefore required number ${=}^{m+n+k}{C}_3{-}^m{C}_3{-}^n{C}_3{-}^k{C}_3$