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Question
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 ______.
Options
`""^((m + n + k))"C"_3`
`""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`
mC3 + nC3 + kC3
mC3 × nC3 × kC3
Solution
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 `""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`.
Explanation:
Here the total number of points are (m + n + k) which must give `""^((m + n + k))"C"_3` number of triangles but m points on l1 taking 3 points at a time gives mC3 combinations which produce no triangle.
Similarly, nC3 and kC3 number of triangles can not be formed.
Therefore, the required number of triangles is `""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`.
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