### Instruction: 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?### Response: I want to count the number of ways to choose a subset of the ten points that form a convex polygon.
I notice that any subset of three or more points will form a convex polygon, since they are all on a circle.
So, I can use the binomial coefficient to count the number of ways to choose k points out of 10, where k is between 3 and 10.
This is the same as adding up the binomial coefficients from 10 choose 3 to 10 choose 10.
Using the formula for the binomial coefficient, I get 10 choose 3 = 10!/(3!7!) = 120, 10 choose 4 = 10!/(4!6!) = 210, and so on.