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  4. Three straight lines l1, l2 and l3 are parallel and lie in the same plane. Five points are taken on each of l1, l2 and l3. The maximum number of triangles which can be obtained with vertices at these points, is

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Q. Three straight lines $l_1, l_2$ and $l_3$ are parallel and lie in the same plane. Five points are taken on each of $l_1$, $l_2$ and $l_3$. The maximum number of triangles which can be obtained with vertices at these points, is

Permutations and Combinations

Solution:

The total number of points is 15 . From these 15 points we can obtain ${ }^{15} C_3$ triangles. However, if all the 3 points are chosen on the same straight line, we do not get a triangle. Therefore, the required number of triangles
$={ }^{15} C_3-3\left({ }^5 C_3\right)=425 .$