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  4. The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers of m points are taken on l1 ; n points on l2, k points on l3. The maximum number of triangles formed with vertices at these points are

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Q. The straight lines $l_1, l_2$ and $l_3$ are parallel and lie in the same plane. A total numbers of $m$ points are taken on $l_1 ; n$ points on $l_2, k$ points on $l_3$. The maximum number of triangles formed with vertices at these points are

Permutations and Combinations

Solution:

Here, the total number of points are $(m+n+k)$ which must give ${ }^{m+n+k} C_3$ combinations by taking 3 points at a time, but $m$ points lying on $l_1$, therefore taking 3 points at a time gives ${ }^m C_3$ combinations which produces no triangle. Similarly, ${ }^n C_3$ and ${ }^k C_3$ number of triangles cannot be formed. Therefore, the required number of triangles is ${ }^{\left(m+n+k\right)} C_3-{ }^m C_3-{ }^n C_3-{ }^k C_3$.