Ten [points](https://artofproblemsolving.com/wiki/index.php/Point) are marked on a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). How many distinct [convex polygons](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) of three or more sides can be drawn using some (or all) of the ten points as [ vertices](https://artofproblemsolving.com/wiki/index.php/Vertex)?
Any [subset](https://artofproblemsolving.com/wiki/index.php/Subset) of the ten points with three or more members can be made into exactly one such polygon.  Thus, we need to count the number of such subsets.  There are $2^{10} = 1024$ total subsets of a ten-member [set](https://artofproblemsolving.com/wiki/index.php/Set), but of these ${10 \choose 0} = 1$ have 0 members, ${10 \choose 1} = 10$ have 1 member and ${10 \choose 2} = 45$ have 2 members.  Thus the answer is $1024 - 1 - 10 - 45 = \boxed{968}$.
Note ${n \choose 0}+{n \choose 1} + {n \choose 2} + \dots + {n \choose n}$ is equivalent to $2^n$