GCSE Maths Product Rule for Counting Exam Questions

So, you want to test your knowledge on GCSE Maths’ product rule for counting?

As well as being tested as a topic in its own right, it is a really helpful skill for probability questions. 

Consider the small football league explained in question 5. When you listed the matches played, you should have got 6 matches in total:

A vs BA vs CA vs DB vs CB vs DC vs D

With just four teams, writing out the different possibilities isn’t too time consuming. If there were 40 teams, this would be completely unrealistic. So, we use the product rule for counting to speed up this process.

Let’s get stuck in with some exam style questions!

Question 1

Ariana is going to choose a starter and a main course at a restaurant. She can choose from 8 starters and 12 main courses.

a. How many different ways are there of choosing a starter and a main course? (2)

b. Ariana decides she would also like a dessert. There are 5 desserts available. How many ways can Ariana choose a starter, a main and a dessert? (1)

a.  8 Γ— 12 = 96

b.  96 Γ— 5 = 480

Question 2

14 teams play each other in a competition. Each team plays every other team exactly once. Work out the total number of games played. (2)

 14 Γ— 13 Γ· 2 = 91

Question 3

Abby is choosing two dancers to perform a duet. She has 18 dancers to choose from.

a. Abby says there are 306 different ways to select the duet. 
Explain why Abby is wrong. (2)

b. Find the correct number of different ways to select the duet. (1)

a.  18 Γ— 17 = 306
The order in which the dancers are picked doesn’t matter, so Abby should have divided by 2. 

b.  306 Γ· 2 = 153

Question 4

Will must pick three students from the sixth form to be prefects. 

One student must be from year 13 and one student must be from year 12. The third student can be from either year. 

This year, 27 year 13 students and 17 year 12 students have applied to be prefects. How many different ways can Will pick from the students who have applied? (3)

27 Γ— 17 = 459 ways to pick one year 13 and one year 12. 

This leaves 26 year 13s and 16 year 12s, so the final student is chosen from a total of 42 applicants. 

459 Γ— 42 = 19 278

Question 5

Idina is buying a new kitchen. She needs to pick the cabinet doors, the work surface and the floor. She can pick from 19 different cabinet doors and she can pick from 17 different work surfaces. The total number of ways Idina can choose the cabinet doors, the work surface and the floor is 5491. 

How many different floors can Idina choose from? (2)

19 Γ— 17 = 323

5491 Γ· 323 = 17 floors 

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