The Product Rule for Counting

The product rule for counting

Welcome, welcome, welcome to Beyond’s third Monthly Maths Mastery blog! This time, we’re taking a close look at the product rule for counting, where it comes from and how to successfully apply it in your own mathematical meanderings. Are you ready? Of course you are. On the count of three then? One, two, three…

How to calculate number of possible outcomes

Imagine you are in a restaurant (it’s not lockdown, and the menu is somewhat limited!). The menu contains 3 different starters – garlic bread, soup and spaghetti bolognese – and 4 main courses – pizza, lasagne, calzone and risotto. How many different ways are there to choose 1 starter and 1 main course?

We could use a method called systematic listing. This, a little obviously, involves listing all possible outcomes using some sort of system.

For instance, let’s use the first letter of each item to describe it. G for garlic bread, S for soup and so on, with the exception of spaghetti bolognese, which we’ll just call B…

The options are:

GP, GL, GB, GR

SP, SL, SC, SR

BP, BL, BC, BR

We can count these up and find we have a total of 12 combinations.

This is all fine and well, but what do we do when dealing with larger groups? Are we going to endlessly scribble down our possible food choices on a napkin?

No, of course not! We use something called the product rule for counting…

What is the product rule for counting?

One way to link the previous problem with the product rule for counting is to think about it in tree diagram form. The first set of branches on this tree will include the starter options, and so there will be three branches. 

The product rule for counting

Then, the second set of branches include the main course options. Of course, the path you choose will depend on which starter you chose and so we include the four branches for the main courses coming off all three branches representing the starters. 

The product rule for counting

This means, when you draw the other branches for the other starters, that there is a total of 3 × 4 = 12 outcomes, and so there are 12 possible combinations. Simple! We love diagrams here at Beyond. Ten points if you can spot the purposeful mistake 😉…

What does the product rule for counting state?

We can generalise this idea to form the product rule for counting. The product rule for counting tells us that the total number of outcomes for two or more events is found by multiplying the number of outcomes for each event together. It is called the product rule for counting because when we multiply numbers together; this is known as finding the product!

For example,

Euan’s wardrobe contains 5 different coloured ties, and 7 different coloured shirts. How many pairings can he choose?

Event 1 is choosing a tie, and this has 5 outcomes. Event 2 is choosing a shirt, and this has 7 outcomes. The product rule for counting says that the total number of outcomes can be found by multiplying these numbers together.

Number of pairings = 5 × 7 = 35

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Can the product rule be used for more than two events?

The product rule can absolutely be used to find the number of outcomes for any number of events! It’s that good!

For example, 

Charlie is trying to think of a pin code. The code can contain 4 digits and each digit can be chosen from the numbers 0 to 9 inclusive. Given that the code will be an even number and any digit can be repeated, find the total number of pin codes that Charlie could use. 

Let’s think about the digits of his number in turn. 

Digit 1: this could be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Charlie can choose from 10 numbers.

Digit 2: this could be any of the numbers listed above. Charlie can choose from 10 numbers.

Digit 3: this could be any of the numbers listed as digit 1. Charlie can choose from 10 numbers.

Digit 4: the number is going to be even, so this could be 0, 2, 4, 6 or 8. Charlie can choose from 5 numbers.

The total number of pin codes is found by multiplying all of the numbers of outcomes together.

10 × 10 × 10 × 5 = 5000

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What happens if the events involve duplicates?

Occasionally, we will need to reduce the number of outcomes due to duplication. This involves dividing by the number of ways to order those duplicates. 

For example,

A scout leader wants to choose 2 children to survey. There are 20 children in the group. How many possible combinations of children can the scout leader choose?

There are 20 ways to choose the first child to survey. Once that child is chosen, there are 19 left to choose from. 

The means the total number of combinations is 20 × 19 = 380

However, choosing Ali then Ben or choosing Ben then Ali gives the same result. That means we need to divide the total number of combinations by 2 to eliminate the duplicates. 

3800 ÷ 2 = 190 

How simple was that? Level 9s all round please…

Need someone to walk you through it? I know I do…simply click the link below!

How can I practise the product rule for counting?

Now that you know how the product rule for counting works, why don’t you try this worksheet? In the meantime, we’ll see you next month for more maths mastery milestones…

Product Rule for Counting Worksheet - Beyond

Missed last month’s maths mastery? Simply click the image below!

Beyond's Monthly Maths Mastery: Graphing Linear and Quadratic Functions

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