Nine points are drawn on a sheet of paper, as shown in the following figure:

a) In how many ways is it possible to choose three collinear points?

b) In how many ways is it possible to choose four points such that three of them are collinear?
Solution

a) There are eight ways to choose three collinear points as illustrated in the figure below:
![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-28.jpg?height=498&width=1080&top_left_y=230&top_left_x=404)

b) To choose a set of four points containing three collinear points, we can adopt a procedure consisting of the two steps below:

(1) first, choose three collinear points;

(2) with these three collinear points fixed, choose one of the remaining 6 points.

Notice that in this way, it is possible to generate all sets of four points that contain three collinear points. Moreover, each choice of this type produces a set distinct from those produced by the other choices. Now we can count how many choices are possible. From part a) above, we know that there are eight choices in the first step of the procedure. Given a choice in the first step of the procedure, there are six possible choices in the second step. Thus, in total, there are $8 \times 6 = 48$ choices. Therefore, there are 48 sets of four points containing three collinear points.