Problem:
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? (Polygons are distinct unless they have exactly the same vertices.)
Solution:
For 3≤k≤10, each choice of k points will yield a convex polygon with k vertices. Because k points can be chosen from 10 in (k10​) ways, the answer to the problem is
(310​)+(410​)+​⋯+(1010​)=[(010​)+(110​)+⋯+(1010​)]−[(010​)+(110​)+(210​)]=(1+1)10−(1+10+45)=968​​
Query: Where have we used the stipulation that the polygons are convex?
The problems on this page are the property of the MAA's American Mathematics Competitions