Our teacher said this was a tricky question but I do not understand why. If I have a circle that has ten dots on it and I have to make as many polygons using those dots, the answer should be 10. Connecting each dot starting with triangle, to square, all the way to the max of a 10-sided polygon should give me only 10 polygons that I can make. Am I missing something here?
2026-03-27 07:20:07.1774596007
There are ten dots on the sphere, how many polygons can be
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There are several ways to read the question which lead to different answers. One way is to ask how many disjoint polygons you can form simultaneously. This would be the number of triangles in a triangulation of the decagon. Another would be to ask how many different polygons that do not have intersecting sides you can form. In this case there is one for any combination of at least three vertices. Another would be to ask how many different polygons you can form if you allow the sides to intersect. In this case for each set of four vertices there are three, not one, polygon that can be formed. For $n$ vertices you can make $\frac {(n-1)!}2$ polygons this way.