\(n\) points are placed on a circle, and each pair of points is joined by a straight line. The points are chosen so that no three of these lines pass through the same point. Let \(r_{n}\) be the number of regions into which the interior of the circle is divided.

Draw pictures to calculate \(r_{n}\) for some small values of \(n\).

Conjecture a formula for \(r_{n}\) in terms of \(n\).