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There are 20 points on a plane, no 3 are collinear except 4. Find no. of lines formed, and total no. of triangles that can be formed.

ATTEMPT

No of lines will be $$\binom{20}{2}-\binom{4}{2}+1 = 185$$

No of Triangles using only given points will be : $$\binom{20}{3}-\binom{4}{3} = 1140 - 4 = 1136$$

But the question asks for total no of triangles. so we would also consider triangles with some vertices that are not one of the given twenty points.

So how would we calculate them?

I attempted like so:

A triangle would form if three lines intersect. But these three lines should not intersect at same point. SO we subtract 3-line combinations that meet at same point from total no of 3-line combinations.

so it should be $$\binom{185}{3}-\left[16\times\binom{19}{3}+4\times\binom{17}{3}\right]$$

$$= 1038220-18224 = \boxed{1019996}$$

But i seriously doubt if this is correct. Such a small number can't produce an astronomical no. of Triangles! Please help. Thank you.

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EDIT

As pointed by joriki, the answer would depend on position of points, and nothing may be said until that is known. But will my answer hold if no 2 lines are parallel?

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The question can't be meant like that, since the result would depend on the positions of the points: You can make a pair of lines parallel to remove one of those triangles without adding another. So it's probably just a sloppy way of asking for the number of triangles formed by the vertices.

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  • $\begingroup$ Thank you! i get that point that two lines can be parallel and so no triangle would form using the third line (transversal). But if we add a condition that no line is parallel, then will my answer hold or will it still be wrong? Thank you. $\endgroup$
    – Max Payne
    Commented Aug 6, 2015 at 13:09
  • $\begingroup$ @TimKrul: Yes, under that assumption, your answer seems correct. Well counted. $\endgroup$
    – joriki
    Commented Aug 6, 2015 at 13:26
  • $\begingroup$ Thank you! Actually question was framed like "find maximum no of triangles that can be formed." $\endgroup$
    – Max Payne
    Commented Aug 7, 2015 at 0:43
  • $\begingroup$ @joriki No i meant in the book it was written maximum. I asked a different question. I didn't mean to ask maximum. $\endgroup$
    – Max Payne
    Commented Aug 7, 2015 at 10:28
  • $\begingroup$ @TimKrul: I see -- apologies for my harsh remark; I've deleted it. I hope you'll forgive me for jumping to that conclusion, since there unfortunately really are so many sloppily asked questions on the site that waste a lot of time and effort. $\endgroup$
    – joriki
    Commented Aug 8, 2015 at 5:12

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