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Seven points are marked on the circumference of a circle and all pairs of points are joined by straight lines. No three of these lines have a common point and any two intersect at a point inside the circle. Into how many regions is the interior of the circle divided by these lines?
Each new chord divides existing regions according to the number of earlier intersection points on it. For seven points, the total number of regions is $$1+\binom{7}{2}+\binom{7}{4}$$. This equals $$1+21+35=57$$.
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