Question

# 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 -

^{m + n + k}C_{3}
^{m + n +} ^{k}C_{3} – ^{m}C_{3} – ^{n}C_{3} – ^{k}C_{3}
^{m}C_{3} + ^{n}C_{3} + ^{k}C_{3}
- None of these

^{m + n + k}C_{3}^{m + n +}^{k}C_{3}–^{m}C_{3}–^{n}C_{3}–^{k}C_{3}^{m}C_{3}+^{n}C_{3}+^{k}C_{3}## The correct answer is: ^{m + n +} ^{k}C_{3} – ^{m}C_{3} – ^{n}C_{3} – ^{k}C_{3}

### Total number of points = m +n + k. Therefore the total number of triangles formed by these points is ^{m + n + k}C_{3}. But out of these m + n + k points, m points lie on I_{1}, n points lie on I_{2} and k points lie on I_{3} and by joining three points on the same line we do not obtain a triangle. Hence the total number of triangles is

^{m + n +} ^{k}C_{3} – ^{m}C_{3} – ^{n}C_{3} – ^{k}C_{3}.

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The coefficient of in is