The straight lines \[{{I}_{1}},\ {{I}_{2}},\ {{I}_{3}}\] are parallel and lie in the same plane. A total number 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 are [IIT Screening 1993; UPSEAT 2001]

\[^{m}{{C}_{3}}{{+}^{n}}{{C}_{3}}{{+}^{k}}{{C}_{3}}\]

\[^{m+n+k}{{C}_{3}}{{-}^{m}}{{C}_{3}}{{-}^{n}}{{C}_{3}}-{{}^{k}}{{C}_{3}}\]

The straight lines \[{{I}_{1}},\ {{I}_{2}},\ {{I}_{3}}\] are parallel

1.	The straight lines \[{{I}_{1}},\ {{I}_{2}},\ {{I}_{3}}\] are parallel and lie in the same plane. A total number 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 are [IIT Screening 1993; UPSEAT 2001]
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» C. \[^{m}{{C}_{3}}{{+}^{n}}{{C}_{3}}{{+}^{k}}{{C}_{3}}\]

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