Q. The straight lines $I_{1},I_{2},I_{3}$ are parallel and lie in the same plane. A total numbers of $m$ points are taken on $I_{1},n$ points on $I_{2},k$ points on $I_{3}$. The maximum number of triangles formed with vertices at these points is
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 the above
Explanation
(b)
Total number of points are $m + n + k$, the triangles formed by these points $= \ ^{m + n + k}C_{3}\$
Joining of three points on the same line gives no triangle, the number of such triangles is $\ ^{m}C_{3} + \ ^{n}C_{3} + \ ^{k}C_{3}$
$\therefore$ Required number of triangles
$=^{m + n + k}C_{3} -^{m}C_{3} -^{n}C_{3} -^{k}C_{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.