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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 L1.n points on L2 and k points on L3. Then maximum number of triangles formed with vertices at these points are

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m + n + k C_{3}

^{n + m + k} C_{3} - (^{n} C_{3} +^{m} C_{3} +^{k} C_{3})

(^{m} C_{3} +^{n} C_{2} +^{k} C_{3})

^{m} C_{2} \cdot^{n} C_{2} +^{n} C_{1} \cdot^{k} C_{2} +^{n} C_{2} \cdot^{m} C_{1}

answer is B.

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Detailed Solution

Straight lines given in the problem are

L 1, L 2, L 3

These lines are parallel as well as lie in the same plane.
Here, total number of points taken on L1 = m
Here, total number of points taken on L2 = n
Here, total number of points taken on L3 = k
Thus, total number of points

= m + n + k

,
For a triangle there are minimum three points needed to be selected.
Hence, the number of ways by which 3 point can be selected,

=^{n + m + k} C_{3}

.
Now, when a form a triangle then all the three points do not lie on the same line.
Thus, three point which lie on the same plane can be given as,

=^{n} C_{3} +^{m} C_{3} +^{k} C_{3}

.
Thus, the maximum number of triangles,

=^{n + m + k} C_{3} - (^{n} C_{3} +^{m} C_{3} +^{k} C_{3}) .

Option 2 is correct.

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