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Jan 30, 2014 at 17:05 comment added Joseph O'Rourke The maximum number of simple polygonizations (that's a key search phrase) for a set of $n$ points has been heavily studied, and is somewhere between $4^n$ and $56^n$. See this link. I don't know of a result for random points.
Jan 30, 2014 at 16:48 history asked kaleidoscop CC BY-SA 3.0