The product rule Quiz (Grade 6)

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Score: 0 / 10

The product rule — Notes & Worked Examples Show / Hide

Explanation:

The product rule for counting states that if there are m ways to make a first choice and n ways to make a second choice, the total number of combinations is m × n.

If items cannot be repeated, reduce the number of available choices by 1 for each next position. If you are choosing a pair of items where order does not matter (e.g. picking a team of 2 people or handshakes), divide by 2 at the end so each pair is not counted twice.

Example 1: A restaurant menu has 5 starters, 8 main courses, and 4 desserts. How many different 3-course meals can be chosen?

Solution:

Multiply the number of choices for each course:

Total meals = 5 × 8 × 4

5 × 8 = 40; 40 × 4 = 160

Answer: 160 meals

Example 2: A 4-digit PIN uses digits from 1 to 9 with no repeated digits. The first digit must be even (2, 4, 6 or 8). How many PINs are possible?

Solution:

1st digit: 4 choices (2, 4, 6, or 8)

2nd digit: 8 remaining digits (from the 9 digits 1–9)

3rd digit: 7 choices; 4th digit: 6 choices

Total = 4 × 8 × 7 × 6 = 1344

Answer: 1344 PINs

One Page Guide (PDF) The product rule Worksheets

Question 1

[2 marks]

Harris is buying a shirt and a tie.
He has a choice of three colours of shirt and a choice of three styles of tie.

Harris is going to choose one shirt and one tie.
List all the possible combinations Harris can choose.

Worked Solution & Answer

3 × 3 = 9 combinations: WP, WS, WC, BP, BS, BC, GP, GS, GC M1 C1

Answer: WP, WS, WC, BP, BS, BC, GP, GS, GC

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Question 2 IGCSE Jun 2021 2F Q7

[2 marks]

Hassan is going to eat at a restaurant.
Here is the menu at the restaurant.

Hassan is going to choose one starter and one main course from the menu.
List all the possible combinations that Hassan can choose.

Worked Solution & Answer

3 starters (F, P, S) × 4 mains (B, C, L, R) = 12 combinations M1 A1

Answer: FB, FC, FL, FR, PB, PC, PL, PR, SB, SC, SL, SR

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Question 3

[2 marks]

Jack is in a restaurant.
There are 6 starters, 8 main courses and some desserts on the menu.
Jack is going to choose one starter, one main course and one dessert.
He says there are 240 ways that he can choose his starter, his main course and his dessert.
Could Jack be correct?
You must show how you get your answer.

Worked Solution & Answer

6 × 8 = 48 M1
240 ÷ 48 = 5 desserts (an integer), so yes Jack could be correct. A1

Answer: Yes — 240 ÷ (6 × 8) = 240 ÷ 48 = 5, which is a whole number of desserts.

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Question 4 AQA Jun 2025 3H Q15

[3 marks]

Amy and Becky each make integers using **three** single digits.
In any integer, digits may be repeated.
• Amy makes **even** integers with a first digit greater than 7
• Becky makes **odd** integers with a first digit that is non-zero.
They each make as many different integers as possible.

How many **more** integers than Amy does Becky make?

Answer

Worked Solution & Answer

Amy: 1st digit ∈ {8, 9} (2 choices), 2nd digit (10 choices), 3rd digit even (5 choices) ⇒ 2 × 10 × 5 = 100 M1
Becky: 1st digit ∈ {1..9} (9 choices), 2nd digit (10 choices), 3rd digit odd (5 choices) ⇒ 9 × 10 × 5 = 450 M1
Difference = 450 – 100 = 350. A1

Answer: Answer 350

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Question 5

[3 marks]

There are 15 dogs in a dog show.
One dog is awarded first prize.
A different dog is awarded second prize.
There are 12 cats in a cat show.
One cat is awarded first prize.
A different cat is awarded second prize.

Work out how many different ways these four prizes can be awarded.

Worked Solution & Answer

Ways for dogs = 15 × 14 = 210 M1
Ways for cats = 12 × 11 = 132 M1
Total ways = 210 × 132 = 27 720. A1

Answer: 27 720

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Question 6 AQA Jun 2025 3F Q3

[3 marks]

Jodie makes a sandwich using one type of bread and one filling.
@@TABULAR₀@@

(a) Complete the table of the 6 possible sandwiches they could make.
One has been done for you. [2 marks]

(b) What fraction of the possible sandwiches use Cheese? [1 mark]

Answer

Worked Solution & Answer

(a)

White bread: WC, WH, WT M1
Brown bread: BC, BH, BT A1

Answer (a): WC, WH, WT, BC, BH, BT

(b)

2 out of the 6 sandwiches contain cheese (WC and BC) ⇒ 2/6 = 1/3 B1

Answer (b): Answer 2/6 (or 1/3)

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Question 7

[3 marks]

Tracey is going to choose a main course and a dessert in a cafe.
She can choose from 8 main courses and 7 desserts.
Tracey says that to work out the number of different ways of choosing a main course and a dessert you add 8 and 7

(a) Is Tracey correct?
You must give a reason for your answer. [1 mark]

(b) Work out the total number of games played. [2 marks]

Worked Solution & Answer

(a)

Product rule: 8 × 7 = 56 C1

Answer (a): No, she should multiply 8 by 7 (giving 56 ways, not 15)

(b)

(12 × 11) ÷ 2 = 66 M1 A1

Answer (b): 66

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Question 8

[4 marks]

There are three dials on a combination lock.
Each dial can be set to one of the numbers 1, 2, 3, 4, 5
The three digit number 553 is one way the dials can be set, as shown in the diagram.

(a) Work out the number of different three digit numbers that can be set for the combination lock. [2 marks]

(b) How many of the possible three digit numbers have three different digits? [2 marks]

Worked Solution & Answer

(a)

5 × 5 × 5 = 125 M1 A1

Answer (a): 125

(b)

5 × 4 × 3 = 60 M1 A1

Answer (b): 60

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Question 9 OCR Jun 2022 4H Q13

[4 marks]

Here is a restaurant’s menu.

(a) A 3-course meal consists of one starter, one main and one dessert.
Work out how many different 3-course meals can be chosen from the menu. [2 marks]

(b) Find the fraction of the 3-course meals which are completely vegetarian (V). [2 marks]

Worked Solution & Answer

(a)

5 starters × 8 mains × 7 desserts = 280. M1 A1

Answer (a): 280

(b)

Vegetarian meals = 2 × 3 × 2 = 12 M1
Fraction = 12/280 = 3/70. A1

Answer (b): 12280 (or 370)

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Question 10 Specimen Set 2 2H Q12

[5 marks]

Marie has 25 cards.
Each card has a different symbol on it.
Marie gives one card to Shelley and one card to Pauline.

(a) In how many different ways can Marie do this? [2 marks]

(b) How many different lists can David write? [3 marks]

Worked Solution & Answer

(a)

25 × 24 = 600 M1 A1

Answer (a): 600

(b)

(boy, girl, boy) = 12 × 10 × 11 = 1320 M1
(girl, boy, girl) = 10 × 12 × 9 = 1080 M1
Total = 1320 + 1080 = 2400 A1

Answer (b): 2400

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Score: 0 / 10 (0 / 31 marks)

The product rule Worksheets Next: Repeated change →

Grade 6 Quizzes

14 Topics

6A - Recurring decimals	6B - Harder indices
6C - The product rule	6D - Repeated change
6E - Triple brackets	6F - Parallel and perpendicular
6G - Inequalities on graphs	6H - Similar shapes (area and vol)
6J - Negative scale factors	6K - Circle theorems
6L - Cumulative frequency	6M - Box plots
6N - Combined transformations	6P - Capture recapture