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The question is :

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?

My answer is 32 . But the actual answer is 31 how ??

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In general the maximum number of regions you can get from $n$ points is given by

$${n \choose 4} + {n \choose 2} + 1$$

This can be proved using induction (other combinatorial proofs exist too). For more information (including at least two proofs), see this: Dividing a circle into areas.

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 $31$.

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