Video: Finding the Number of Ways to Choose 𝑛 out of 𝑚 Things

Video Transcript

Find the number of ways to select three different letters from the set 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑘, 𝑗.

Well we’ve got eight different letters that we can choose from, and we want to select three of them.

Now given that they’re all different letters, when I come to choose my first letter, I’ve got a choice of eight that I can choose from. So in the first box, I’ve got eight. Now when I’ve picked one letter, I can’t choose that one again because they’ve all got to be different letters, so I’ve now only got seven different ones to choose from. So on the second box, I’ve got a choice of seven letters. And having picked out those two letters, I’ve now only got six to choose from in the last box. Now each of those eight ways for the first box can be combined with each of those seven ways for the second box, and they can be combined with each of the six ways for the last box.

So that means we’ve got eight times seven times six, which is 336 different ways. But that’s not quite the answer, Let’s say we’ve picked 𝑐 for the first box and 𝑑 for the second box and 𝑒 for the third box. We might have chosen the letters 𝑒 and 𝑑 in a different order, or maybe we picked 𝑑 first, or even 𝑒.

So within the 336 ways of organizing our three letters, chosen from those eight letters, each group of three has been represented in six different orders. So it’s the same group of three letters represented six different ways, so we’ve kind of multiply counted some of our combinations.

So given that I’ve not specified that I’m interested in the order that the letters come out and I’m gonna group all those six different combinations there as the same group of three letters, I’m only going to have a sixth as many ways of picking out three letters from that group of eight. So the total number of ways is 336 divided by six, which is 56.

And that’s our answer. But before we finish, let’s just talk about some alternative notation. We have a general formula for this sort of problem. How many ways to choose 𝑟 different objects from a set of 𝑛 different objects? And we call that an 𝑛 choose 𝑟 type of problem.

Now depending on where you live, you’ll have seen one of these ways of expressing that particular formula: 𝑛𝐶𝑟, 𝐶𝑛,𝑟, 𝐶(𝑛,𝑟), 𝐶𝑛𝑟, 𝑛𝐶𝑟, or just 𝑛 over 𝑟 in kind of a vector format. But they all boil down to this basic formula: 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial.

Now in our question we were choosing from eight different objects, so 𝑛 was equal to eight. And we were selecting three of them, so 𝑟 is equal to three.

So our calculation becomes eight factorial over three factorial times eight minus three factorial. Now eight minus three is five, so that becomes five factorial on the denominator. Now if we think about what factorial means, so eight factorial means eight times seven times six times five times four times three times two times one; three factorial is three times two times one; and five factorial is five times four times three times two times one.

Now we can do some cancelling; we’ve got one on the top, one on the bottom; two on the top, two on the bottom; three on the top, three on the bottom; four on the top, four on the bottom; five on the top, and five on the bottom. And now we got back to eight times seven times six over three times two times one.

And that’s the 336 over six that we talked about when we did the problem the first way. So this method also gives us the same correct answer: 56.