Question
The straight lines $${l_1},{l_2},{l_3}$$ are parallel and lie in the same plane. A total number of $$m$$ points are taken on $${l_1} , n$$ points on $${l_2} , k$$ point on $$l_3 .$$ 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.
$$^m{C_3} + {\,^m}{C_3} + {\,^k}{C_3}$$
D.
None of these
Answer :
$$^{m + n + k}{C_3} - {\,^m}{C_3} - {\,^n}{C_3} - {\,^k}{C_3}$$
Solution :
The straight line $${l_1},{l_2},{l_3}$$ are parallel and lie in the same plane.
Total number of points $$= m + n + k$$
Total no. of triangles formed with vertices $$ = {\,^{m + n + k}}{C_3}$$
By joining three given points on the same line we don’t obtain a triangle.
Therefore, the max. number of triangles $$ = {\,^{m + n + k}}{C_3} - {\,^m}{C_3} - {\,^n}{C_3} - {\,^k}{C_3}$$