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Can someone please illustrate how this plane works by drawing. I just want to get an idea of how this plane works.

There are n points in the plane, such that no three points are on the same line. a) How many straight lines contain a pair of these points? b) How many triangles contain three of these points as vertices? Any help would be great!

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See also here for your first question:… – Martin Sleziak Apr 13 '14 at 5:19
Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Martin Sleziak Apr 13 '14 at 5:20

Given $n$ non-collinear points, there exists a distinct line for every pair of points and hence there are $\dbinom{n}2$ lines through these $n$ points.

Similarly, a triangle is uniquely determined by $3$ non-collinear points and hence the number of triangles is $\dbinom{n}3$.

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User141421 has answered your question. I just want to mention that one can easily construct for any $n$, points satisfying your hypothesis. For example, choose any set of distinct points in the unit circle. So you can, for some mangeable size of $n$, construct and see.

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