There are nine points on a piece of paper. No three of the points are collinear. How many different triangles can be formed by using three of the nine points as vertices?

Answer 1

Number of possible choices for the first point = 9.

Number of possible choices for the second point = 8.

Number of possible choices for the third point = 7.

Possible triangles with specific order of points = 9*8*7 = 504.

However, when we say triangle, we are not really concerned in which order the points are selected. So we have over-counted the number of triangles by 6, which is the number of ways to order three points.

The number of distinct triangles is therefore 504/6=84.

This number is mathematically called
9 choose 3, calculated by
9!/(3!(9-3)!) = 84
where 9! is factorial 9, = 9*8*7*...*2*1