Question

Consider an overlapping set of four circles A, B, C, D. One would like to position the circles so that every possible subset of the circles forms a region, e.g., four regions each contained in just one (different) circle, six regions formed by the intersection of two circles (AB, AC, AD, BC, BD, CD), four regions formed by the intersection of three of the four circles, and one region formed by the intersection of all four circles. Prove that it is not possible to have such a set of 15 bounded regions.

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