AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 15

3 marks · Medium difficulty · Multi-step Problem

Calculate how many more three-digit integers Becky can make compared to Amy using the product rule for counting with given restrictions on the digits.

Practise this question

Question

Question 15 asks: Amy and Becky each make integers using three single digits. In any integer, digits may be repeated. Amy makes even integers with a first digit greater than 7. Becky makes odd integers with a first digit that is non-zero. They each make as many different integers as possible. How many more integers than Amy does Becky make? Worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 15: M1 for (Amy) 2 multiplied by 10 multiplied by 5 with at least one correct or 100 (must be a product of three integers). M1 for (Becky) 9 multiplied by 10 multiplied by 5 with at least one correct or 450 (must be a product of three integers). A1 for final answer of 350. Additional guidance notes: 2 + 10 + 5 is M0, and use of fractions, decimals or percentages is M0 unless recovered.

How to answer it

Product Rule for Counting: Making 3-Digit Integers

📋 What this question tests
  • Applying the product rule for counting (combinatorics) to multi-digit numbers.
  • Understanding place value restrictions for three-digit integers (first, middle, and last digits).
  • Recognising conditions for even numbers (ending in 0, 2, 4, 6, 8) and odd numbers (ending in 1, 3, 5, 7, 9).
  • Handling constraints such as "greater than 7", "non-zero", and repeated digits.

Question 15 (3 Marks)

Amy and Becky make 3-digit integers with specific constraints

💡 Key Knowledge

  • Single digits available: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} — there are 10 total digits.
  • Repetition allowed: The choice for each position does not reduce the number of options available for other positions.
  • Product Rule: Total combinations = (choices for digit 1) × (choices for digit 2) × (choices for digit 3).
  • Even vs Odd:
    • Even integers must end in: 0, 2, 4, 6, 8 (5 options).
    • Odd integers must end in: 1, 3, 5, 7, 9 (5 options).

🧠 Exam Technique: "Slot" Method

Draw three empty boxes or lines representing the three digits: [_] [_] [_]

  1. Fill restricted slots first: Look at conditions on the 1st digit and 3rd digit.
  2. Check the middle slot: With no conditions, any of the 10 digits (0 to 9) can be used.
  3. Multiply the possibilities: Always multiply independent choices together; never add them.
  4. Read the final demand carefully: The question asks "How many more...?", which requires a final subtraction.

📐 Step-by-Step Calculation

Step 1: Calculate Amy's total integers

  • 1st digit: Greater than 7 → must be 8 or 9 = 2 options
  • 2nd digit: Any single digit (0–9) = 10 options
  • 3rd digit: Even integer → must be 0, 2, 4, 6, or 8 = 5 options
  • Amy's total = 2 × 10 × 5 = 100
awarded M1 for: 2 × 10 × 5 (with at least one correct digit count) or stating 100

Step 2: Calculate Becky's total integers

  • 1st digit: Non-zero → digits 1 to 9 = 9 options
  • 2nd digit: Any single digit (0–9) = 10 options
  • 3rd digit: Odd integer → must be 1, 3, 5, 7, or 9 = 5 options
  • Becky's total = 9 × 10 × 5 = 450
awarded M1 for: 9 × 10 × 5 (with at least one correct digit count) or stating 450

Step 3: Find the difference

  • Difference = Becky − Amy = 450 − 100 = 350
awarded A1 for correct final answer of 350

✅ Final Answer

350

Becky makes 350 more integers than Amy.

❌ Common Errors & Examiner Traps

  • Adding instead of multiplying: Writing 2 + 10 + 5 = 17 scores M0. Independent possibilities across slots must be multiplied.
  • Miscounting the middle digit: Thinking there are only 9 digits between 0 and 9. Remember: 0 through 9 is 10 digits!
  • "Greater than 7" trap: Including 7 (thinking 7, 8, 9 = 3 choices). The word greater is strict, so only 8 and 9 are valid.
  • Forgetting 0 is even: Thinking even digits are only {2, 4, 6, 8} (4 options) instead of {0, 2, 4, 6, 8} (5 options).
  • Stopping early: Calculating 100 and 450 correctly but forgetting to subtract them to answer "How many more...?".

Topics

Number · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.