Skip to content

Paper Together: sit a paper in a room with your friends.See how it works

The product rule for counting practice questions with answers

Higher · Grade 6 · Non-calculator

A second set of the product rule for counting questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

Sue packs 5 shirts, 2 trousers and 2 hats. Calculate the number of outfits.

Show answer

Answer: 20

  1. Multiply independent choices: 5 × 2 = 10.
  2. Multiply by hats: 10 × 2 = 20.
  • Method (1): Multiply independent choices: 5 × 2 = 10.
  • Accuracy (1): Multiply by hats: 10 × 2 = 20.

Question 2 Easy

Tom packs 6 shirts, 1 pair of trousers and 3 hats. Calculate the number of outfits.

Show answer

Answer: 18

  1. Multiply independent choices: 6 × 1 = 6.
  2. Multiply by hats: 6 × 3 = 18.
  • Method (1): Multiply independent choices: 6 × 1 = 6.
  • Accuracy (1): Multiply by hats: 6 × 3 = 18.

Question 3 Medium

Wes picks a captain and a vice-captain from 6 people. Calculate the number of ways.

Show answer

Answer: 30

  1. 6 choices for captain, 5 remain for vice-captain.
  2. Multiply: 6 × 5 = 30.
  • Method (1): 6 choices for captain, 5 remain for vice-captain.
  • Accuracy (1): Multiply: 6 × 5 = 30.

Question 4 Medium

Yan picks a captain and a vice-captain from 10 people. Calculate the number of ways.

Show answer

Answer: 90

  1. 10 choices for captain, 9 remain for vice-captain.
  2. Multiply: 10 × 9 = 90.
  • Method (1): 10 choices for captain, 9 remain for vice-captain.
  • Accuracy (1): Multiply: 10 × 9 = 90.

Question 5 Hard

Choose 2 people from 4 when order does not matter. Calculate the number of ways.

Show answer

Answer: 6

  1. Calculate ordered pairs: 4 × 3 = 12.
  2. Divide by 2 for unordered: 12 / 2 = 6.
  • Method (1): Calculate ordered pairs: 4 × 3 = 12.
  • Accuracy (1): Divide by 2 for unordered: 12 / 2 = 6.

Question 6 Hard

Choose 2 people from 8 when order does not matter. Calculate the number of ways.

Show answer

Answer: 28

  1. Calculate ordered pairs: 8 × 7 = 56.
  2. Divide by 2 for unordered: 56 / 2 = 28.
  • Method (1): Calculate ordered pairs: 8 × 7 = 56.
  • Accuracy (1): Divide by 2 for unordered: 56 / 2 = 28.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Dot packs 3 shirts, 4 trousers and 3 hats. Calculate the number of possible outfits.

Show answer

Answer: 36

  1. The choices are independent, so multiply the counts.
  2. 3 × 4 × 3 = 36 outfits.
  • Method (1): Multiplies the three counts, for example 3 × 4 = 12
  • Accuracy (1): Total 36 outfits

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

Why multiply shirt, trouser and hat counts?

Show answer

Each item from one list pairs with every item from the others. Each item from one list pairs with every item from the others.

Check 2

Outfit count for 3 shirts, 2 trousers and 4 hats?

Show answer

3 × 2 × 4 = 24 outfits. 3 × 2 × 4 = 24 outfits.

Check 3

Outfit count for 2 shirts, 5 trousers and 3 hats?

Show answer

2 × 5 × 3 = 30 outfits. 2 × 5 × 3 = 30 outfits.

Check 4

Menu product: 4 choices and 5 choices equals

Show answer

4 × 5 = 20 choices. 4 × 5 = 20 choices.

Check 5

Menu product: 4 choices and 7 choices equals

Show answer

4 × 7 = 28 choices. 4 × 7 = 28 choices.

Check 6

Why is a captain and a vice-captain from 8 people equal to 56?

Show answer

The second choice has 7 people left. The second choice has 7 people left.

Check 7

What is a captain and a vice-captain from 6 people?

Show answer

6 × 5 = 30 ways. 6 × 5 = 30 ways.

Check 8

What is a captain and a vice-captain from 5 people?

Show answer

5 × 4 = 20 ways. 5 × 4 = 20 ways.

Check 9

Why divide by 2 when order does not matter?

Show answer

Each pair was listed twice in the ordered product. Each pair was listed twice in the ordered product.

Check 10

How many ways can you choose 2 people from 5 when order does not matter?

  1. 21 ways
  2. 15 ways
  3. 10 ways
Show answer

10 ways 5 × 4 ÷ 2 = 10 ways.

Check 11

How many ways can you choose 2 people from 6 when order does not matter?

  1. 10 ways
  2. 21 ways
  3. 15 ways
Show answer

15 ways 6 × 5 ÷ 2 = 15 ways.

Check 12

How many ways can you choose 2 people from 7 when order does not matter?

  1. 15 ways
  2. 21 ways
  3. 10 ways
Show answer

21 ways 7 × 6 ÷ 2 = 21 ways.

Check 13

Why is 8×8 wrong for captain and vice-captain from 8?

Show answer

One person cannot hold both jobs. One person cannot hold both jobs.

Check 14

A single trouser option: 6 shirts, 1 pair of trousers and 3 hats gives how many outfits?

Show answer

6 × 1 × 3 = 18 outfits. 6 × 1 × 3 = 18 outfits.

Check 15

What should you decide before halving a product?

Show answer

Whether the order of the two people matters. Whether the order of the two people matters.

Keep going with The product rule for counting

Practise free

Start free

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Next topicPractise free