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I am reading through "Proofs: A Long-Form Mathematics Textbook" and they say "suppose you were investigating how many regions are formed if one places n dots randomly on a circle and then connects them with lines." They give the examples n = 1: 1 region, n = 2: 2 regions, n = 3: 4 regions, n = 4: 8 regions and finally n = 5: 16 regions. My questions come when he says the general formula for this is (some long quadratic polynomial that is not really relevant). If we are assuming n dots are placed randomly, who is to say (however unlikely) that these dots could not always stack on each other? If this was the case then it would be 0 regions created each time, giving us a "general formula" of 0.

What am I missing here?

tothemax
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    It is implied that they all land in different spots. – Randall Jan 11 '23 at 00:46
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    Not only in different spots, but in general position, which is to say that there are no "coincidental" triple points (such as if $6$ points were equally spaced around the circle so that the center is a single intersection for all three diameters). – Sammy Black Jan 11 '23 at 01:33

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