3-3. On a circle, points \(A_{1}, A_{2}, \ldots, A_{16}\) are given. We will construct all possible convex polygons whose vertices are among the points \(A_{1}, A_{2}, \ldots, A_{16}\). We will divide these polygons into two groups. The first group will include all polygons where \(A_{1}\) is a vertex. The second group will include all polygons where \(A_{1}\) is not a vertex. In which group are there more polygons?
Solve 3. Answer: in the second. Each polygon not containing point \(A_{1}\) can be associated with a polygon containing point \(A_{1}\) by simply adding \(A_{1}\) to its vertices. The reverse operation (removing vertex \(A_{1}\)) is only possible for polygons with no fewer than 4 vertices.