Given two parallel lines, the first one containing N points and the second containing M points. How many triangles and convex quadrilaterals can be formed if no line contains more than 1000 points

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Given two parallel lines, the first one containing    N  points and the second containing    M  points. How many triangles and convex quadrilaterals can be formed if no line contains more than    1000  points

geoiste72 · Accepted Answer

Hint: Do triangles and quads separately.I&#039;m assuming that degenerate triangles (all three verts on one line) aren&#039;t counted. So every triangle must have   2 verts on one line, and one on the other.How many ways are there of picking   2 verts on the first line?   N  (  N  −  1  )      /    2. And for each of these, you can pick any vert on the other line. So for triangles like this, you have   M  N  (  N  −  1  )      /    2. For two verts on the other line and one vert on the first, you have   N  M  (  M  −  1  )      /    2. Total  M  N  (  N  +  M  −  2  )      /    2A similar analysis lets you work out quads, but you have to be careful: pick two verts from the first line, IN ORDER; then pick two verts from the second line, ALSO IN ORDER. Assuming that a degenerate quad (one with three or four verts on one line) isn&#039;t allowed, you get   N  M  (  N  −  1  )  (  M  −  1  )      /    4 ways to pick a quad, I believe.Sounds as if the problem might have needed a little more detail to indicate whether degenerate triangles/quads are allowed.

Given two parallel lines, the first one containing N points and the second containing M points. How many triangles and convex quadrilaterals can be formed if no line contains more than 1000 points

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