There are $\mathbf{n}$ lines in a plane no two of which are parallel. They intersect at $^{n}C_2$ distinct points in space. How many new lines are constructed by joining these points.

The essential thing to note in this question is the term $\mathbf{'new lines'}$ as several of the lines formed by joining the points will simply be coincident with the original lines and as such $\binom{^{n}C_2}{2}$ will not be the answer.


Let $p_1$ and $p_2$ be points of intersection of distinct lines $l_1,l_2$ and distinct lines $l_3,l_4$ respectively. The line passing through $p_1$ and $p_2$ (note that $p_2$ and $p_1$ are distinct) is a 'new line' iff $\{l_1,l_2,l_3,l_4\}$ contains four distinct elements(lines). Hence, there can be a maximum of $\frac{1}{2}\binom{n}{2}\binom{n-2}{2}$ newlines.

Although, any two lines intersect at distinct points, there is no guarantee that each unordered pair of distinct intersection points gives a unique 'new line' as the picture below suggests. (intersection points A, D G are on the same line.)

enter image description here

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