Formulas for max number of intersection points on a plane, and in space

Question

The maximum number of intersection points of $n$ lines on a plane is given by $n\choose2$. How do we derive this formula? I see that we can $n\choose2$ $= \dfrac{n(n-1)}{2}$, where the $rhs$ is just the sum of $1+2+...+n$.
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and also, how do we get that the maximum number of vertices formed in $R^3$ by $n$ planes is given by $n\choose3$. I see that we can only form a vertex in $R^3$ with $3$ planes, but not sure how to derive the formula.

Thanks

Answer 1

Two lines can intersect each other once or not at all. An intersection point is formed by a pair of two lines. We get the maximum intersection points by counting all possible pairs of two lines. There are ${n \choose 2}$ such pairs and since they can only form one point each it's the maximum of intersection points. The same argument works for the planes aswell.

I hope this might help you