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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Number

Systematic Listing and the Product Rule

How many different meals, codes or arrangements are possible? Foundation questions ask you to list them carefully. Higher questions use the product rule to count without listing. Both come down to being organised.

1Listing outcomes systematically

To make sure you list every combination exactly once, fix the first choice and run through all the second choices, then move to the next first choice.

Worked example
A café sells tea (T) or coffee (C), with cake (K), biscuit (B) or fruit (F). List all the possible combinations.
  1. Fix T: TK, TB, TF.
  2. Fix C: CK, CB, CF.
6 combinations: TK, TB, TF, CK, CB, CF
Exam tipAlways use a pattern: alphabetical, or smallest first. Then you can check you haven’t missed one.

2Listing arrangements

Worked example
List all the 3-digit numbers that can be made using each of the digits 2, 5 and 8 once.
  1. Starting with 2: 258, 285.
  2. Starting with 5: 528, 582.
  3. Starting with 8: 825, 852.
6 numbers

Notice: 3 choices for the first digit, then 2, then 1, so there are 3 × 2 × 1 = 6 arrangements. This leads to the product rule.

3The product rule for counting

Product ruleIf there are m ways to do one thing and n ways to do another, there are m × n ways to do both. This extends to any number of choices.
Worked example
A menu has 3 starters, 4 mains and 2 desserts. How many different three-course meals are there?
  1. 3 × 4 × 2.
24 meals
Worked example
How many 4-digit PINs are possible (a) if digits can repeat (b) if no digit can repeat?
  1. (a) 10 choices for each digit: 10 × 10 × 10 × 10 = 10 000.
  2. (b) 10 choices, then 9, then 8, then 7: 10 × 9 × 8 × 7 = 5040.
(a) 10 000 (b) 5040

4Arrangements of n different objects

The number of ways to arrange n different objects in a line is n × (n − 1) × … × 2 × 1. This is written n! (“n factorial”).

Worked example
How many ways can the letters of the word MATHS be arranged?
  1. 5 different letters: 5 × 4 × 3 × 2 × 1.
120

5Counting with restrictions

Deal with the restricted positions first, then fill the rest.

Worked example
How many even 3-digit numbers are there (100 to 999)?
  1. Last digit (restricted): 0, 2, 4, 6 or 8 → 5 choices.
  2. First digit: 1–9 → 9 choices.
  3. Middle digit: 0–9 → 10 choices.
  4. 9 × 10 × 5.
450
Worked example
5 people sit in a row. A and B must sit at the two ends. How many arrangements are there?
  1. The ends: A then B, or B then A → 2 ways.
  2. The other 3 people fill the middle seats: 3 × 2 × 1 = 6 ways.
  3. 2 × 6.
12

6Choosing pairs

When order doesn’t matter, such as handshakes or matches, the product rule counts each pair twice, so divide by 2.

Worked example
6 people each shake hands once with everyone else. How many handshakes are there?
  1. Each of the 6 people shakes 5 hands: 6 × 5 = 30.
  2. Each handshake has been counted twice: 30 ÷ 2.
15 handshakes
PairsNumber of pairs from n items = n(n − 1)2
Common mistakes
  • Listing without a system and missing or repeating combinations.
  • Adding instead of multiplying: 3 starters and 4 mains give 3 × 4 = 12 meals, not 7.
  • Forgetting that the first digit of a 3-digit number can’t be 0.
  • Counting handshakes or matches twice. Divide by 2 when order doesn’t matter.
  • Not reducing the choices when items can’t be repeated.

★Key Vocabulary Recap

Systematic
Done in an ordered way so nothing is missed.
Product rule
Multiply the number of choices at each stage.
Factorial (n!)
n × (n − 1) × … × 1.
Arrangement
An ordering of items where order matters.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: N5 (systematic listing; product rule for counting — Higher))
  • N5: apply systematic listing strategies.
  • N5 (Higher): including use of the product rule for counting.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Number

Systematic Listing and the Product Rule — Practice Questions

12 exam-style questions (27 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation · Grades 2–3
  1. (2 marks)
    Grade 2
    A café sells tea or coffee, with cake, biscuit or fruit. List all the possible combinations of one drink and one snack.
  2. (2 marks)
    Grade 2
    List all the 3-digit numbers that can be made using each of the digits 2, 5 and 8 exactly once.
  3. (1 mark)
    Grade 3
    A coin is flipped and an ordinary six-sided dice is rolled. How many possible outcomes are there?
Foundation and Higher · Grades 4–5
  1. (2 marks)
    Grade 4
    A restaurant offers 4 starters, 6 main courses and 3 desserts. How many different three-course meals are there?
  2. (2 marks)
    Grade 5
    A 4-digit PIN uses the digits 0–9. No digit may be used more than once. How many different PINs are possible?
  3. (2 marks)
    Grade 5
    A code is made of 2 letters (A–Z) followed by 3 digits (0–9). Letters and digits may repeat. How many codes are possible?
Higher · Grades 6–7
  1. (2 marks)
    Grade 6
    Two fair dice are rolled. How many of the outcomes give a total of 7? Hence find the probability of a total of 7.
  2. (3 marks)
    Grade 6
    How many even numbers are there between 100 and 999 inclusive?
  3. (2 marks)
    Grade 6
    In how many different ways can the letters of the word MATHS be arranged?
  4. (3 marks)
    Grade 7
    Five friends, including Amir and Bea, sit in a row of five seats. Amir and Bea must sit in the two end seats. How many different seating arrangements are there?
  5. (3 marks)
    Grade 7
    In a league of 8 teams, every team plays every other team once. How many matches are played?
  6. (3 marks)
    Grade 7
    How many 3-digit numbers greater than 500 have three different digits?
    HintStart with the first digit. Can it be 5? Is 500 itself allowed?
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Number

Systematic Listing and the Product Rule — Worked Solutions

Full worked solutions to all 12 questions (27 marks), set out the way an Edexcel examiner wants to see them.

Foundation · Grades 2–3
  1. (2 marks)
    Grade 2
    A café sells tea or coffee, with cake, biscuit or fruit. List all the possible combinations of one drink and one snack.
    Solution
    1. Tea: cake, biscuit, fruit.
    2. Coffee: cake, biscuit, fruit.
    6 combinations
  2. (2 marks)
    Grade 2
    List all the 3-digit numbers that can be made using each of the digits 2, 5 and 8 exactly once.
    Solution
    1. 258, 285, 528, 582, 825, 852.
    6 numbers
  3. (1 mark)
    Grade 3
    A coin is flipped and an ordinary six-sided dice is rolled. How many possible outcomes are there?
    Solution
    1. 2 × 6.
    12
Foundation and Higher · Grades 4–5
  1. (2 marks)
    Grade 4
    A restaurant offers 4 starters, 6 main courses and 3 desserts. How many different three-course meals are there?
    Solution
    1. 4 × 6 × 3.
    72
  2. (2 marks)
    Grade 5
    A 4-digit PIN uses the digits 0–9. No digit may be used more than once. How many different PINs are possible?
    Solution
    1. 10 × 9 × 8 × 7.
    5040
  3. (2 marks)
    Grade 5
    A code is made of 2 letters (A–Z) followed by 3 digits (0–9). Letters and digits may repeat. How many codes are possible?
    Solution
    1. 26 × 26 × 10 × 10 × 10.
    676 000
Higher · Grades 6–7
  1. (2 marks)
    Grade 6
    Two fair dice are rolled. How many of the outcomes give a total of 7? Hence find the probability of a total of 7.
    Solution
    1. Pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes.
    2. Total outcomes 6 × 6 = 36, so P = 636 = 16.
    6; 16
  2. (3 marks)
    Grade 6
    How many even numbers are there between 100 and 999 inclusive?
    Solution
    1. First digit 1–9: 9 choices. Middle: 10 choices. Last (even): 5 choices.
    2. 9 × 10 × 5.
    450
  3. (2 marks)
    Grade 6
    In how many different ways can the letters of the word MATHS be arranged?
    Solution
    1. 5 different letters: 5 × 4 × 3 × 2 × 1.
    120
  4. (3 marks)
    Grade 7
    Five friends, including Amir and Bea, sit in a row of five seats. Amir and Bea must sit in the two end seats. How many different seating arrangements are there?
    Solution
    1. End seats: 2 ways (Amir left or Bea left).
    2. Middle three seats: 3 × 2 × 1 = 6 ways.
    3. 2 × 6.
    12
  5. (3 marks)
    Grade 7
    In a league of 8 teams, every team plays every other team once. How many matches are played?
    Solution
    1. 8 × 7 = 56 counts each match twice.
    2. 56 ÷ 2.
    28
  6. (3 marks)
    Grade 7
    How many 3-digit numbers greater than 500 have three different digits?
    Solution
    1. First digit 5, 6, 7, 8 or 9: 5 choices (500 itself has a repeated digit, so it isn’t counted).
    2. Second digit: any of the 9 digits not already used.
    3. Third digit: 8 remaining choices.
    4. 5 × 9 × 8.
    360
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