Suppose that three of the labelled points are chosen at random. What is the probability that the chosen points are collinear?
Well, first let’s just check what collinear means. It means that the points lie in a straight line. So we’ve got five points to choose from. We’re gonna choose three of them at random. What’s the probability that those three points all lie on a straight line?
Now looking at those five points, the only three which all lie on the same straight line are 𝐹, 𝐺, and 𝐻. Now it doesn’t matter whether we pick 𝐻 first then 𝐺 then 𝐹 or 𝐹 then 𝐺 then 𝐻 or any other order of those three points, but they’re the three points that we must pick in order to get collinear points.
So we’ve got five different points to choose from, and we need to choose three of them. So we’re gonna use our 𝑛-choose-𝑟 formula where 𝑛, the number of different points, is five and 𝑟, the number that we need to choose, is three.
Now there are a lot of different ways of notating that particular formula, so you choose the one that’s familiar to you. But they all boil down to this calculation: 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. So if we just plug our numbers in, 𝑛 is five and 𝑟 is three, we get five factorial over three factorial times five minus three factorial. Well five minus three is two.
So this simplifies to five factorial over three factorial two factorial. And of course five factorial means five times four times three times two times one; three factorial means three times two times one; and two factorial is just two times one. Now we can do a bit of cancelling The threes, the twos, and the ones there cancel. So I’ve got five times four over two times one, which is 20 over two, which is 10.
So there are 10 ways of choosing three different points from our five different points. But only one of those ways involves having 𝐻, 𝐺, and 𝐹 as the three letters that we’re interested in.
So out of the 10 different ways that we’ve got of selecting three letters from five only, one of them is that the group 𝐹, 𝐺, 𝐻. So that means only one out of the 10 possible outcomes results in the chosen points being collinear, so the answer is the probability that they are collinear is one over 10, a tenth.