The Product Rule for Counting

What you'll learn

How to count choices without listing every possibility.
When to multiply the number of options.
How to deal with unknown numbers of choices, like $x$ drinks.
When to divide by 2 for pairs, matches and handshakes.

1. Choices, stages and outcomes

In counting questions, you often make a decision in parts.

A stage is one part of a decision, such as choosing a sandwich first, then choosing a drink.

Definition

Options and outcomes

An option is one possible choice at a single stage.
An outcome is one complete result after all stages have been chosen.

For example, if you choose one sandwich and one drink, then “cheese sandwich with orange juice” is one complete outcome.

Example

Listing first to see the pattern

A lunch offer has 2 wraps and 3 drinks. How many different lunches are possible?

Each of the 2 wraps can be paired with each of the 3 drinks, giving 6 complete lunches.

Start with the first wrap. It can go with any of the 3 drinks.
Start with the second wrap. It can also go with any of the 3 drinks.
So there are 3 outcomes for the first wrap and 3 outcomes for the second wrap.
The total number of lunches is 6.

Listing is fine for small numbers, but it gets slow very quickly. That is where the product rule helps.

2. The product rule

Definition

The product rule for counting

If a task has one stage with $a$ options and another stage with $b$ options, and every option from the first stage can go with every option from the second stage, then the number of outcomes is $\times b$ .

Key Idea

One from each group means multiply

When you choose one item from each separate group, multiply the number of choices in each group.

Example

Choosing one boy and one girl

A club has 11 boys and 13 girls. One boy and one girl are chosen for a presentation. How many different pairs are possible?

Choosing one boy and one girl means every boy can be paired with every girl.

There are 11 choices for the boy.
For each boy, there are 13 choices for the girl.
Multiply the number of choices:
$11 \times 13 = 143$
There are 143 possible boy-and-girl pairs.

Common Mistake

Adding instead of multiplying

Do not do 11 + 13 here. Adding would only tell you how many students there are altogether, not how many boy-and-girl pairs can be made.

3. More than two stages

The product rule works for more than two choices as well. If you choose a starter, then a main course, then a dessert, that is 3 stages.

You multiply all the stage counts together.

Example

Choosing a three-course meal

A restaurant offers 4 starters, 6 main courses and 3 desserts. How many different three-course meals can be chosen?

A three-course meal is made in three stages: starter, main course, then dessert.

There are 4 choices for the starter.
For each starter, there are 6 choices for the main course.
For each starter-and-main combination, there are 3 choices for dessert.
Multiply all three numbers:
$\times 6 \times 3 = 72$
There are 72 possible three-course meals.

Tip

Read the word “and” carefully

In these questions, “choose a starter and a main and a dessert” usually means one from each group, so you multiply the choices.

Common Mistake

Check all combinations are allowed

The product rule works directly when every option at one stage can go with every option at the next stage. If some combinations are not allowed, you may need to subtract or count cases separately.

4. Questions with an unknown number of choices

Sometimes a question gives the total number of outcomes and asks whether a claim could be correct.

If the unknown number of choices is written as $x$ , set up a multiplication equation using the product rule.

A number of choices must be a whole number, such as 0, 1, 2, 3 and so on. It cannot be a decimal.

Example

Checking whether a claim could be correct

A meal deal has 8 sandwich choices and $x$ drink choices. Sam says there are 90 possible meal deals. Could Sam be correct?

The total number of meal deals is represented by an 8 by x array of sandwich-and-drink combinations.

Use the product rule:
$\times x = 90$
Find $x$ by dividing 90 by 8:
$\div 8 = 11.25$
The number of drink choices must be a whole number.
Since 11.25 drinks is impossible, Sam cannot be correct.

Common Mistake

Stopping after the division

In “could this be correct?” questions, you must say whether the value of $x$ makes sense. A decimal number of people, drinks or desserts is not possible.

5. When choices are used up

Some counting questions involve choosing from the same group more than once.

For example, if you deal cards from a deck, once a card has been given out, it cannot be chosen again.

Definition

Without replacement

Without replacement means that once an item has been chosen, it is not put back, so there is one fewer option next time.

Example

Giving cards to two named people

A full deck has 52 cards. One card is given to Maya and one card is given to Leo. How many different ways can this happen?

After Maya receives one card, only 51 cards remain for Leo.

Maya can receive any one of the 52 cards.
After Maya has received a card, there are 51 cards left for Leo.
Multiply the choices:
$52 \times 51 = 2652$
There are 2652 different ways.

Here, the people are named. Maya getting the ace and Leo getting the king is different from Maya getting the king and Leo getting the ace.

Common Mistake

Forgetting the choice has gone

Do not use 52 choices for both people. The second person only has 51 choices because one card has already been given away.

6. When order does not matter: pairs

Sometimes you are just choosing a pair, with no first person and second person.

An unordered pair is a pair where the order does not matter. For example, Alex and Ben is the same pair as Ben and Alex.

If there are $n$ items and you choose 2 of them as a pair, the product rule first gives $n (n - 1)$ , but that counts every pair twice.

So the number of pairs is:

n(n−1)2\frac{n(n-1)}{2}

Key Idea

Pairs are counted twice

For pairs, matches and handshakes, divide by 2 because each pair can be listed in two orders.

Example

Choosing two students for the same prize

There are 28 students in a year group. Two students are chosen to receive the same prize. How many different pairs could be chosen?

Because the prize is the same, the pair Alex–Ben is identical to Ben–Alex, so ordered choices are counted twice.

If the students were chosen as first and second, there would be 28 choices, then 27 choices.
Multiply and then divide by 2 because the order does not matter:
$28×272=378\frac{28 \times 27}{2} = 378$
There are 378 possible pairs of students.

The same idea applies to teams playing matches and people shaking hands. A match between Team A and Team B is the same match as Team B against Team A.

Example

Counting games between teams

There are 9 teams in a league. Each team plays every other team once. How many games are played?

Each game is one unordered connection between two teams, so Team A vs Team B is the same as Team B vs Team A.

Choose the first team in 9 ways.
Choose the second team in 8 ways.
This counts each game twice, so divide by 2:
$9×82=36\frac{9 \times 8}{2} = 36$
There are 36 games in total.

Common Mistake

Counting both directions as new games

For ordinary matches or handshakes, Team A against Team B is not a different event from Team B against Team A. Divide by 2 unless the roles are different.

Exam technique

In the exam

Identify the stages: what is being chosen first, second, third?
Multiply when you need one choice from each separate group.
If items are chosen from the same group, reduce the number of options each time.
Ask whether order matters. If you are just making pairs, divide by 2.
If there is an unknown like $x$ , solve for it and check it is a whole number.

Self review

Check yourself

Why do you multiply for a meal with one starter and one main course?
When giving two cards to two named people, why does the second person have fewer choices?
Why do handshake questions usually involve dividing by 2?

The Product Rule for Counting

What you'll learn

How to count choices without listing every possibility.
When to multiply the number of options.
How to deal with unknown numbers of choices, like $x$ drinks.
When to divide by 2 for pairs, matches and handshakes.

1. Choices, stages and outcomes

In counting questions, you often make a decision in parts.

A stage is one part of a decision, such as choosing a sandwich first, then choosing a drink.

Definition

Options and outcomes

An option is one possible choice at a single stage.
An outcome is one complete result after all stages have been chosen.

For example, if you choose one sandwich and one drink, then “cheese sandwich with orange juice” is one complete outcome.

Example

Listing first to see the pattern

A lunch offer has 2 wraps and 3 drinks. How many different lunches are possible?

Each of the 2 wraps can be paired with each of the 3 drinks, giving 6 complete lunches.

Start with the first wrap. It can go with any of the 3 drinks.
Start with the second wrap. It can also go with any of the 3 drinks.
So there are 3 outcomes for the first wrap and 3 outcomes for the second wrap.
The total number of lunches is 6.

Listing is fine for small numbers, but it gets slow very quickly. That is where the product rule helps.

2. The product rule

Definition

The product rule for counting

Key Idea

One from each group means multiply

When you choose one item from each separate group, multiply the number of choices in each group.

Example

Choosing one boy and one girl

A club has 11 boys and 13 girls. One boy and one girl are chosen for a presentation. How many different pairs are possible?

Choosing one boy and one girl means every boy can be paired with every girl.

There are 11 choices for the boy.
For each boy, there are 13 choices for the girl.
Multiply the number of choices:
$11 \times 13 = 143$
There are 143 possible boy-and-girl pairs.

Common Mistake

Adding instead of multiplying

Do not do 11 + 13 here. Adding would only tell you how many students there are altogether, not how many boy-and-girl pairs can be made.

3. More than two stages

The product rule works for more than two choices as well. If you choose a starter, then a main course, then a dessert, that is 3 stages.

You multiply all the stage counts together.

Example

Choosing a three-course meal

A restaurant offers 4 starters, 6 main courses and 3 desserts. How many different three-course meals can be chosen?

A three-course meal is made in three stages: starter, main course, then dessert.

There are 4 choices for the starter.
For each starter, there are 6 choices for the main course.
For each starter-and-main combination, there are 3 choices for dessert.
Multiply all three numbers:
$\times 6 \times 3 = 72$
There are 72 possible three-course meals.

Tip

Read the word “and” carefully

In these questions, “choose a starter and a main and a dessert” usually means one from each group, so you multiply the choices.

Common Mistake

Check all combinations are allowed

The product rule works directly when every option at one stage can go with every option at the next stage. If some combinations are not allowed, you may need to subtract or count cases separately.

4. Questions with an unknown number of choices

Sometimes a question gives the total number of outcomes and asks whether a claim could be correct.

If the unknown number of choices is written as $x$ , set up a multiplication equation using the product rule.

A number of choices must be a whole number, such as 0, 1, 2, 3 and so on. It cannot be a decimal.

Example

Checking whether a claim could be correct

A meal deal has 8 sandwich choices and $x$ drink choices. Sam says there are 90 possible meal deals. Could Sam be correct?

The total number of meal deals is represented by an 8 by x array of sandwich-and-drink combinations.

Use the product rule:
$\times x = 90$
Find $x$ by dividing 90 by 8:
$\div 8 = 11.25$
The number of drink choices must be a whole number.
Since 11.25 drinks is impossible, Sam cannot be correct.

Common Mistake

Stopping after the division

In “could this be correct?” questions, you must say whether the value of $x$ makes sense. A decimal number of people, drinks or desserts is not possible.

5. When choices are used up

Some counting questions involve choosing from the same group more than once.

For example, if you deal cards from a deck, once a card has been given out, it cannot be chosen again.

Definition

Without replacement

Without replacement means that once an item has been chosen, it is not put back, so there is one fewer option next time.

Example

Giving cards to two named people

A full deck has 52 cards. One card is given to Maya and one card is given to Leo. How many different ways can this happen?

After Maya receives one card, only 51 cards remain for Leo.

Maya can receive any one of the 52 cards.
After Maya has received a card, there are 51 cards left for Leo.
Multiply the choices:
$52 \times 51 = 2652$
There are 2652 different ways.

Here, the people are named. Maya getting the ace and Leo getting the king is different from Maya getting the king and Leo getting the ace.

Common Mistake

Forgetting the choice has gone

Do not use 52 choices for both people. The second person only has 51 choices because one card has already been given away.

6. When order does not matter: pairs

Sometimes you are just choosing a pair, with no first person and second person.

An unordered pair is a pair where the order does not matter. For example, Alex and Ben is the same pair as Ben and Alex.

If there are $n$ items and you choose 2 of them as a pair, the product rule first gives $n (n - 1)$ , but that counts every pair twice.

So the number of pairs is:

n(n−1)2\frac{n(n-1)}{2}

Key Idea

Pairs are counted twice

For pairs, matches and handshakes, divide by 2 because each pair can be listed in two orders.

Example

Choosing two students for the same prize

There are 28 students in a year group. Two students are chosen to receive the same prize. How many different pairs could be chosen?

Because the prize is the same, the pair Alex–Ben is identical to Ben–Alex, so ordered choices are counted twice.

If the students were chosen as first and second, there would be 28 choices, then 27 choices.
Multiply and then divide by 2 because the order does not matter:
$28×272=378\frac{28 \times 27}{2} = 378$
There are 378 possible pairs of students.

The same idea applies to teams playing matches and people shaking hands. A match between Team A and Team B is the same match as Team B against Team A.

Example

Counting games between teams

There are 9 teams in a league. Each team plays every other team once. How many games are played?

Each game is one unordered connection between two teams, so Team A vs Team B is the same as Team B vs Team A.

Choose the first team in 9 ways.
Choose the second team in 8 ways.
This counts each game twice, so divide by 2:
$9×82=36\frac{9 \times 8}{2} = 36$
There are 36 games in total.

Common Mistake

Counting both directions as new games

For ordinary matches or handshakes, Team A against Team B is not a different event from Team B against Team A. Divide by 2 unless the roles are different.

Exam technique

In the exam

Identify the stages: what is being chosen first, second, third?
Multiply when you need one choice from each separate group.
If items are chosen from the same group, reduce the number of options each time.
Ask whether order matters. If you are just making pairs, divide by 2.
If there is an unknown like $x$ , solve for it and check it is a whole number.

Self review

Check yourself

Why do you multiply for a meal with one starter and one main course?
When giving two cards to two named people, why does the second person have fewer choices?
Why do handshake questions usually involve dividing by 2?

The Product Rule for Counting

What you'll learn

1. Choices, stages and outcomes

2. The product rule

3. More than two stages

4. Questions with an unknown number of choices

5. When choices are used up

6. When order does not matter: pairs

Recap questions

You've reached the end

The Product Rule for Counting

What you'll learn

1. Choices, stages and outcomes

2. The product rule

3. More than two stages

4. Questions with an unknown number of choices

5. When choices are used up

6. When order does not matter: pairs

Recap questions

You've reached the end

The Product Rule for Counting

What you'll learn

1. Choices, stages and outcomes

2. The product rule

3. More than two stages

4. Questions with an unknown number of choices

5. When choices are used up

6. When order does not matter: pairs

Recap questions

You've reached the end

The Product Rule for Counting Revision Guide

The Product Rule for Counting

What you'll learn

1. Choices, stages and outcomes

2. The product rule

3. More than two stages

4. Questions with an unknown number of choices

5. When choices are used up

6. When order does not matter: pairs

Recap questions

You've reached the end

The Product Rule for Counting Revision Guide