Question

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?

Answer

Answer: 968

Concepts:combinations complementary counting subsets

Difficulty rating: 1780

Small Hint:

Each choice of at least three marked points determines one convex polygon

Big Hint:

Count all subsets and remove those of sizes $0,$ $1,$ and $2$

Solution:

Every subset of at least three points determines exactly one convex polygon. There are $2^{10}=1024$ subsets in all. Of these, $\binom{10}{0}=1,$ $\binom{10}{1}=10,$ and $\binom{10}{2}=45$ have fewer than three points. Therefore the required number is $1024-1-10-45=968.$

1989 AIME Problem 2

Answer: 968

Difficulty rating: 1780

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years