Tree diagram related question?

Question

I have a question that goes like this:

10 points are marked on a straight line, and 11 points are marked on another line parallel to the first one. How many:
a) quadrilaterals
b) triangles
are there with vertices at these points?

Answer 1

You have 21 points, and you can get both solutions using techniques from combinatorics.

a) A quadrilateral has 4 points. So first, determine how many ways you can choose 4 points from 21. Now, eliminate all of the cases where 3 or 4 points are chosen from one line, because that eliminates a vertex. So if C(n, r) is the number of ways to choose r items from a set of n items, your answer is C(21, 4) - (C(10, 3) + C(10, 4) + C(11, 3) + C(11, 4)).

b) This solution is very similar, but simpler. The answer is C(21, 3) - (C(10, 3) + C(11, 3)). Here, you only need to eliminate the instances where all 3 points occur in one line.