Product Rule for Counting
Master product rule for counting for GCSE Maths with structured, exam-style practice. This Higher resource covers using the product rule for counting and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Multiply the number of choices at each stage.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
The product rule for counting says that if one task can be done in \(m\) ways and another in \(n\) ways, the two together can be done in \(m \times n\) ways.
So three shirts and four pairs of trousers give \(12\) different outfits. The rule extends to any number of stages, multiplying the options at each one.
The key question is whether repetition is allowed. A four-digit code with repetition has \(10^4\) possibilities, but if no digit may repeat there are \(10 \times 9 \times 8 \times 7\), because each choice removes one option from the next.
Revision notes
The basic rule
Multiply the number of choices at each stage.
With \(5\) starters and \(6\) mains, there are \(30\) possible two-course meals.
Repetition allowed
If items can repeat, the number of options stays the same at each stage.
A three-digit code using digits \(0\) to \(9\) has \(10 \times 10 \times 10 = 1000\) possibilities.
Repetition not allowed
Each choice reduces the options for the next.
Arranging \(4\) books on a shelf gives \(4 \times 3 \times 2 \times 1 = 24\) orders, since each book used is no longer available.
Key points
- Multiply the choices at each stage.
- \(m\) ways then \(n\) ways gives \(m \times n\).
- The rule extends to any number of stages.
- With repetition, the options stay the same each time.
- Without repetition, the options reduce by one each time.
- Check carefully whether repetition is allowed.
Worked examples
Example 1
There are \(4\) shirts and \(7\) ties. How many combinations are possible?
Working
Example 2
A \(3\)-digit code uses digits \(0\) to \(9\) with repetition. How many codes are possible?
Working
Example 3
In how many orders can \(5\) people stand in a line?
Working
Common mistakes
Adding instead of multiplying.
Combinations across stages multiply; adding counts only single choices.
Ignoring whether repetition is allowed.
It changes the calculation completely.
Forgetting the options reduce without repetition.
Each item used is no longer available for later stages.
Missing a stage.
Count the stages carefully before multiplying.
Exam tips
- Write out the number of options at each stage first.
- Check whether items may repeat.
- Reduce the count by one at each stage if they may not.
- Sense-check that the total is plausible.
Key terms
- Product rule
- Multiplying the choices at each stage.
- Repetition
- Whether an item may be used more than once.
- Arrangement
- An ordering of items.
- Combination
- A selection of items.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.