GCSE Maths: Product Rule for Counting

So, how do we figure out the number of combinations of different things? Well, let me show you a quick example. If I let's say we're going to a shop and you can get a main course of chicken or beef, let's say it's a burger or something, and a choice of three drinks. How many combinations are there in total? Well, you might expect that there's, you know, chicken plus coke. Then there's chicken plus water. Then there's chicken plus lemonade. Then there's beef plus coke. Then there's beef plus water. Then there's beef plus lemonade. So, in total, we have six combinations. But there's actually a much faster way to work it out because what if we had five different mains and 10 different drinks? Do we really have to add them all up? The answer to that is no. And this is really useful in in many cases. Instead, here's a method that I use to work it out. Instead, I draw two little boxes. The boxes represent our choices. So, for example, here it's going to be main and drink. And all you do in the boxes, you write the number of options you have. So, for example, with the main, I have two options. So, I'm going to write two in there. And with the drink, I'm going to write that there's three options. And then all you do is you multiply the numbers together to get six possibilities. And that's the really quick way to do it. So, I'm going to show you a few different examples now. And where it applies. This applies to a lot of different types of questions and scenarios. So, the first one is there are 12 boys and 15 girls in a class. One boy and one girl will be selected to represent the class on the school council. Work out the total number of ways of choosing a boy and a girl. Well, again, let's say that's for boys and this is for girls. There are 12 different boys. We're going to put 12 here. And there are 15 different girls. And then all we have to do is multiply those two numbers together. And that's it, which would be 180. Again, let me show you another example here. There are three different choices to make. There's a starter, main course, and dessert. So again, all we're going to do is we're going to put a big box for each of these. We have starters, main courses, and desserts. There are four starters, so stick a four in the front. There are seven main courses, and then there are four desserts. And then all we have to do is multiply all three numbers together. So, let's see. 4 * 7 is 28. And 28 * 4. So, we double it 56. Double again 12. See how quick and easy this is? This one's a fairly interesting thing. Uh, there are 52 cards in a deck. Peter's going to give one card to Casper and one to Kelly. How many different ways are there of doing this? Now, this is a bit interesting because we have to actually think a little bit. So, there are two combinations, right? Two choices, which is the first and the second card. Now, how many cards can we choose from in the first case? Well, there are 52 cards. So, I have 52 cards to choose from. Right? Now, if I've given Peter's given one of these cards to Casper, how many cards are left for the second person for Kelly? Well, that means there's going to be one less card to give. So, that' be 51. And then you just times those two together. That's it. So, it's still the exact same thing. Now, here's an interesting case because sometimes it's not as simple as what I've made it out to be. So, it says there are 30 students in a class. Two students are going to be selected to receive a prize. How many different pairs of students can be picked? Now, here's the total number of pairs of students that can be picked. Again, there are 30 options for the first choice and there are 29 options for the second choice, right? So, all we have to do is do 30 * 29. So, 30 * 30 is 900. So, minus 30 be 870. Now, why did I highlight the different pairs? Well, if you notice, let's go through an example really quick. Let's let's just pick let's say five students, right? So say Tom, Jack, Emma, Priya, and I don't know, John. Now, if I write out these combinations, here's the combinations that we have. So, we have Tom and Jack, Tom and Emma, Tom and Priya, Tom and John, then we have Jack and Tom. You have Jack and Emma. Then you have Jack and Priya. Jack and Jon. I don't know if you've spotted what the problem is here yet. Uh, so let's do one more. So with Emma, we have Emma and Tom. Emma and Jack, Emma and Priya, Emma and John. Now, if you notice, it says different pairs of students, but look at these combinations. Tom and Jack, and Jack and Tom are the same combination. Tom and Emma and Emma and Tom are the same combination. Tom and Priya and actually we didn't do Pria so that one it will be replaced again but you know uh Tom and John again we haven't done John's yet but we have Jack and Emma same same uh Jack and Priya we didn't do Pria so she hasn't got one yet and so on. So if you notice half of these combinations are repeated right? So, I don't need all of those combinations. And if you were to write out every single combination, you'd see that half of them are duplicates. So, when it says something like different pairs or something along those lines, implying that you're not counting every single combination, but you're counting the different possibilities, then all you have to do at the end is half your number. So, hopefully that makes a bit of sense. That'd be what? 485. So, I give you another example. There are eight teams in a combination combination in a competition. Each team plays every other team once. So we're not double counting here, right? Because it's it the implication is different pairs of teams that are playing each other. Work out the total number of games played. So eight teams in a combination. Again, if you use the same logic, if we had team A through E or whatever, well E isn't actually the eighth letter, but whatever, you'd see that they get double counted. So again, we're going to do 8 * 7 because there are eight possibilities for the first team, seven possibilities for the second. And then all you're going to do is you're going to half it afterwards. So again, that would be 56 and then divided by two means there's 28 total number of games played. Now I'm going to show you some other situations which will hopefully make things a little easier for you. Here for the next one, there are 10 people in a room. Each person shakes each other person's hand once. Again, implying that it has to be different combinations. So again, for the first person and the second person, right? Because two people shake hands, right? Well, there are 10 options for the first person, nine options for the second. So we're going to multiply those two numbers together to get 90. And then we're going to divide by two because again, we're we want unique solutions here, right? So person A shaking person's B hand and B shaking A's is the same thing. So we're not counting it. So we're halfing the number. Again, this gets really important when you get into statistics as you get older as well. And lastly, let's do one last question because this is a really common scenario they give you, which is a combination lock, working out how many different possibilities there can be. So each dial can be set to one of the numbers 1 2 3 4 or 5. Work out the number of three-digit numbers. So again, now if you notice, it does say different three-digit numbers. It doesn't mention that each digit has to be different. It's just the overall number. So how many options are there for the first number? Well, it can be 1 2 3 four or five. So it's five options. Now for the next one, there's also five options. And for the one after that, there's also five options. So all we're doing here is we're doing 5 * 5 * 5, which is 125. Now part B of the question could be something like, how many combinations could there be if each digit has to be different? Well, then all you'd have to do is there are still five possibilities for the first number, but there's only four possibilities for the second number, and there's only three possibilities for the last number. So 4 * 3 is 12. 12 * 5 is 60. So it actually halfves the number of combinations. So again, hopefully that helps out with a product rule for counting. Always think about if you're looking for different combinations in the sense that you know each combination has to be completely unique. So, for example, something like this or like this.