Seven points are marked on the line, and one point is marked outside this line
Seven points are marked on the line, and one point is marked outside this line. Find the number of all possible triangles with vertices at the eight marked points.
All triangles will have one of their vertices at a point outside the straight line. Let’s count how many sides of triangles can be obtained by connecting points lying on a straight line. Each of the 7 points can be connected to 6 others. This will turn out 7 6 = 42 segments. But at the same time, we counted each segment twice (point 1 with point 2 and 2 with 1 are the same segment). This means that there will be 21 segments on a straight line, which can be a side of a triangle. By connecting the ends of each line segment to a point outside the straight line, we get 21 triangles.