GCSE & IGCSE maths practice

Product rule for counting questions and answers

If there are $m$ ways to do one thing and $n$ ways to do another, there are $m \times n$ ways to do both. Multiply the number of choices at each stage. If items can't be repeated, each stage has one fewer choice; if a position has a restriction, fill that position first.

8 questions · Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 27 Oct 2026.

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Where product rule for counting comes up

GCSE (AQA, Edexcel, OCR)	Foundation and Higher: systematic listing strategies. Higher only: the product rule for counting.
Cambridge IGCSE (0580)	Not in the 0580 syllabus.

The method

Count the choices for each stage.
Multiply them.
No repeats: one fewer choice each time.
Restriction (e.g. must be even): fill that position first.

Product rule for counting questions

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Warm-up

Question 1Warm-upHigher

A menu has 3 starters and 4 mains. How many different two-course meals are there?

Question 2Warm-upHigher

The menu adds 2 desserts. How many three-course meals are there?

Exam-style

Question 3Exam-styleHigher

How many 4-digit PINs can be made using the digits 0 to 9 (repeats allowed)?

Question 4Exam-styleHigher

In how many different orders can the letters of MATHS be arranged?

Question 5Exam-styleHigher

How many 3-digit codes use the digits 0 to 9 with no digit repeated?

Question 6Exam-styleHigher

A team of 11 chooses a captain and a vice-captain (different people). How many ways can this be done?

Challenge

Question 7ChallengeHigher

A code is 2 letters (A to Z) followed by 3 digits (0 to 9), with repeats allowed. How many codes are possible?

Question 8ChallengeHigher

How many 3-digit even numbers can be made from the digits 1 to 9, with no digit repeated?

Common mistakes

Adding the choices instead of multiplying.
Forgetting 0 is a digit (0 to 9 is 10 choices).
Filling the restricted position last.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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