Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Number
Practise Listing and the product rule. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Listing all the possible combinations in a systematic way, and using the product rule to count them without listing. Expect 2 to 3 mark questions about meals, codes or arrangements; Higher papers add restrictions such as no repeats, pairs, or conditions on certain positions.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
List outcomes in a systematic orderE.g. the two-digit numbers made from 3, 5 and 8 without repeats: 35, 38, 53, 58, 83, 85.
3
List combinations from two or three setsFix one item and run through every option for the others, then check you have them all.
4
Use the product rule for two choicesIf there are \(m\) ways to do one thing and \(n\) ways to do another, there are \(m \times n\) ways to do both.
5
Use the product rule for several choicesE.g. 3 letters then 2 digits: 26 × 26 × 26 × 10 × 10 = 1,757,600 codes.
Notes
Systematic listing
Work in a fixed order so you don't miss or repeat an outcome.
For combinations from two lists, fix the first item and pair it with every item in the second list, then move on to the next first item.
Example: 3 sandwiches (cheese C, egg E, ham H) and 2 drinks (tea T, juice J) give CT, CJ, ET, EJ, HT, HJ: 6 combinations.
If order doesn't matter (e.g. choosing two flavours), AB and BA are the same choice: list each pair only once.
The product rule
If one choice can be made in \(m\) ways and a second, separate choice in \(n\) ways, both choices can be made in \(m \times n\) ways.
It works for any number of choices: 4 starters, 6 mains and 3 desserts give 4 × 6 × 3 = 72 different meals.
A 4-digit PIN using the digits 0 to 9, with repeats allowed: 10 × 10 × 10 × 10 = 10,000 PINs.
Use the product rule to check a list: 3 sandwiches × 2 drinks = 6 combinations.
Repeats allowed: a 3-digit code from 0 to 9 has 10 × 10 × 10 = 1000 options
List systematically: fix the first item, and change the last item fastest
How to answer each type of question
List all the possible combinations
2 marks3
Use letters or short labels for each item.
Fix the first choice and go through every option for the second, then move on.
Check the number of items in your list with the product rule.
Example. Asha chooses one base and one topping for a pizza. Bases: thin (T), deep (D). Toppings: ham (H), mushroom (M), pepper (P), olive (O). List all the possible combinations she could choose.
Show the model answer
TH, TM, TP, TO, DH, DM, DP, DO (B2 for all 8 with no extras or repeats; B1 for at least 4 correct)
Use the product rule: how many different ways?
2 marks4
Count the number of options for each choice.
Multiply them together (don't add).
Write the multiplication down, then the answer.
Example. A menu has 5 starters, 7 main courses and 4 desserts. Ben chooses one of each. How many different meals could Ben choose?
Show the model answer
5 × 7 × 4 (M1) = 140 (A1)
Shortcuts and memory tricks
Draw a dash for each position (_ _ _ _), write the number of options on each dash, then multiply.
Check a list with the product rule: 2 bases × 4 toppings should give 8 combinations.
'Different' or 'no repeats' means the numbers go down by 1 each time.
Work in a pattern: keep the first item fixed and change the last item fastest.
Where marks are lost
Adding instead of multiplying: 5 starters and 7 mains give 35 meals, not 12.
Listing without a system, and missing or repeating combinations.
Allowing repeats when the question says the letters or digits are different.
Ignoring a condition such as 'the first digit cannot be 0' or 'the number must be even'.
Counting the same pair twice (AB and BA) when the order doesn't matter.
Exam technique
When asked to list, set out the list clearly in order; one missing or repeated item usually costs a mark.
Show the multiplication (e.g. 26 × 25 × 9 × 10 × 10), not just the answer, to earn the method mark.
Underline words such as 'different', 'no repeats', 'even', 'greater than' and 'cannot start with'.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
Isla has three cards. There is a number on each card.\[\boxed{1}\quad\boxed{7}\quad\boxed{9}\]Isla uses all three cards to make 3-digit numbers.
(a) List all the different 3-digit numbers Isla can make.[2]
(b) How many of these numbers are greater than 400?[1]
Show the answer and mark scheme
(a)Answer: 179, 197, 719, 791, 917, 971
B2 for all 6 numbers and no extras: 179, 197, 719, 791, 917, 971
Worked solution: Work systematically, starting with each card in turn: 179, 197, 719, 791, 917, 971
(b)Answer: 4
B1 for 4 (ft their list)
Worked solution: 719, 791, 917, 971 are greater than 400.
Question 2Medium3 marks
A password is made of 3 letters from A to Z followed by 2 digits from 0 to 9. The letters can be repeated. The two digits must be different. Ravi says, 'The number of possible passwords is \(26^3 \times 10^2\).'
(a) Explain the mistake in Ravi's method.[1]
(b) Work out the number of possible passwords.[2]
Show the answer and mark scheme
(a)Answer: \(10^2\) allows the two digits to be the same; once the first digit is chosen there are only 9 choices for the second.
C1 for explaining that there are only 9 choices for the second digit (\(10^2\) includes repeated digits such as 33)
Worked solution: The two digits must be different, so the second digit has 9 choices, not 10.