# 6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?

6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?
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elulamami
In general the maximum number of regions you can get from 𝑛 points is given by $\left(\genfrac{}{}{0}{}{n}{4}\right)+\left(\genfrac{}{}{0}{}{n}{2}\right)+1$
This can be proved using induction (other combinatorial proofs exist too).
This is an oft cited puzzle to show the perils of generalizing based on first few values. We get powers of $2$ till $n=5$, after which we get $31$.