Systematic listing and the product rule - Edexcel GCSE Maths (Non-calculator) Exam Questions

Practise 29 exam-style questions on Systematic listing and the product rule for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

29 questions
1–5 marks each
Instant marking
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20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

Sam has four coins: 10p, 20p, 50p and £1.
He picks two different coins.

List all the possible total amounts of money he could pick.

Model answer

Pair 10p with each larger coin: 10p + 20p = 30p, 10p + 50p = 60p, 10p + £1 = £1.10
Pair 20p with each larger coin: 20p + 50p = 70p, 20p + £1 = £1.20
Pair 50p with £1: 50p + £1 = £1.50
Totals: 30p, 60p, 70p, £1.10, £1.20, £1.50

Explanation

List systematically: pair the smallest coin with each of the others, then the next smallest with each larger coin, and so on. This makes sure you include every pair exactly once (10p + 20p is the same pair as 20p + 10p).

Mark scheme

2 marks
B2 for all six totals: 30p, 60p, £1.10, 70p, £1.20, £1.50 (and no others)
any order; B1 for at least four correct totals with no more than one incorrect or repeated

Question 2

2 marks

There are 3 different roads from town A to town B.
There are 4 different roads from town B to town C.
There are also 2 different roads that go directly from town A to town C without passing through town B.

How many different routes are there from town A to town C?

Model answer

Routes through B: 3 × 4 = 12
Direct routes: 2
Total: 12 + 2 = 14

Explanation

For routes through B, every road from A to B can be followed by any road from B to C, so multiply. The direct roads are separate alternatives, so they are added on. Multiply for 'and then', add for 'or'.

Mark scheme

1 mark
M1 for 3 × 4 (= 12)
correct answer with no working scores full marks
1 mark
A1 for 14
cao

Question 3

3 marks

Students choose one subject from each of three blocks.
Block A has 4 subjects, including French.
Block B has 5 subjects.
Block C has 3 subjects, including Spanish.
Students are not allowed to choose both French and Spanish.

How many different combinations of three subjects can a student choose?

Model answer

Without the rule: 4 × 5 × 3 = 60 combinations
Combinations with both French and Spanish: 1 × 5 × 1 = 5
Allowed combinations: 60 − 5 = 55

Explanation

Count all the combinations with the product rule, then subtract those that break the rule. A combination with French and Spanish can have any of the 5 Block B subjects, so there are 5 to remove.

Mark scheme

1 mark
P1 for 4 × 5 × 3 (= 60)
correct answer with no working scores full marks
1 mark
P1 for the number of combinations with both French and Spanish: 1 × 5 × 1 (= 5)
or counting directly: 3 × 5 × 3 + 1 × 5 × 2 (= 45 + 10)
1 mark
A1 for 55
cao

Question 4

4 marks

Kayla has three 1p coins, two 5p coins and one 20p coin.

How many different amounts of money can she make using one or more of these coins?
You must explain your answer.

Model answer

Number of 1p coins used: 0, 1, 2 or 3 (4 choices)
Number of 5p coins used: 0, 1 or 2 (3 choices)
Number of 20p coins used: 0 or 1 (2 choices)
4 × 3 × 2 = 24 selections, but one of these uses no coins: 24 − 1 = 23
All the amounts are different: the 1p coins give 0 to 3p, the 5p coins add 0, 5 or 10p, so 1p and 5p coins give 0-3, 5-8 or 10-13p (no overlaps), and the 20p adds 20p, giving 20-33p.
So there are 23 different amounts.

Explanation

Count the choices of how many of each type of coin to use, multiply, and remove the selection with no coins. You must also check that two different selections never give the same amount; that is true here because the coins of each value can never add up to the next coin's value.

Mark scheme

1 mark
P1 for 4 choices for the number of 1p coins (0 to 3), 3 for the 5p coins (0 to 2) and 2 for the 20p coin
or a systematic list of at least 15 different amounts
1 mark
P1 for 4 × 3 × 2 (= 24)
1 mark
P1 for subtracting 1 for using no coins
1 mark
A1 for 23 with an explanation that all the amounts are different
e.g. the 1p coins only make up to 3p, less than 5p, and the 1p and 5p coins together make at most 13p, less than 20p

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