Triangles and Lines living together
The straight lines L1, L2, L3 are parallel and lie in the same plane. A total number of 1000 points are taken line L1, 2000 points on line L2, 2000 points on line L3. The maximum number of triangles formed with vertices at these points, are ?
Note: This problem is of my brother who is not on Brilliant. You may use a calculator.
The answer is 17992000000.
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1 solution
Total no. of points = 1000 + 2000 + 2000 = 5000. The triangles formed are (5000)C(3) = 5000!/(3!X4997!) = 20820835000. But three collinear points give no triangle. These no triangles are (1000)C(3) + (2000)C(3 ) + (2000)C(3) = 2662668000 + 166167000. Therefore no. of triangles = 20820835000 - (2662668000 + 166167000) = 17992000000