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

tizamazg

tizamazg

Open question

2022-08-20

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

Answer & Explanation

geoiste72

geoiste72

Beginner2022-08-21Added 4 answers

Hint: Do triangles and quads separately.
I'm assuming that degenerate triangles (all three verts on one line) aren'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 ) / 2
A 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'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.

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