Problem:
Fifteen distinct points are designated on : the vertices , and ; other points on side other points on side ; and other points on side . Find the number of triangles with positive area whose vertices are among these points.
Solution:
There are ways to select distinct vertices from among the points. All of these selections give vertices of a triangle with positive area except for the selections consisting of collinear points. There are ways to select points on side , ways to select points on side , and ways to select points on side . Thus there are triangles with positive area.
The problems on this page are the property of the MAA's American Mathematics Competitions