Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier)

Q: How do I revise Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier) for GCSE?

Read the notes on Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier), then test yourself without looking. Answering questions and checking what you missed builds recall faster than re-reading.

Q: Is this Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier) guide aligned to the Pearson Edexcel specification?

Yes. This guide is linked to the Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier) topic in the Pearson Edexcel GCSE specification.

Systematic listing ensures all possible outcomes are found without repetition or omission.

Study notes

From the checked teaching notes for this specification point.

The idea

Systematic listing ensures all possible outcomes are found without repetition or omission. For example, to find all two-digit numbers from cards 7, 8, and 9, you can fix the first digit and list the second: 78, 79. Then move to the next first digit: 87, 89, and finally 97, 98. This gives 6 combinations. For larger problems, the product rule is a shortcut. If there are m ways for a first choice and n for a second, there are m × n total outcomes. A menu with 4 starters, 5 mains and 3 desserts offers 4 × 5 × 3 = 60 meal combinations. For a 4-digit PIN from 0-9, there are 10⁴ = 10,000 options if digits can repeat, but 10 × 9 × 8 × 7 = 5,040 if they cannot, as each choice reduces the available pool of digits.

What earns the marks

Demonstrating a clear, systematic method to list all outcomes (e.g., using a table, tree diagram, or ordered list).
Identifying the correct number of independent choices available at each stage of a process.
Correctly multiplying the number of choices for each stage to find the total number of outcomes.
Reducing the number of choices at subsequent stages when items are selected without replacement (no repeats).

Common mistakes and how to put them right

Creating a disorganised list which misses outcomes or includes duplicates.
Put it right: Fix the first item and cycle through all options for the second, then change the first item and repeat the process.
Adding the number of options at each stage instead of multiplying them (e.g., 4 + 5 + 3 instead of 4 × 5 × 3 for the menu problem).
Put it right: The product rule applies to sequential choices, so you must multiply.
Forgetting to reduce the number of available options when repeats are not allowed.
Put it right: If an item is used, it cannot be used again, so the pool of options for the next choice is smaller by one.

Exam tips

If a question asks you to 'list all possible combinations' and the number is small, use a systematic approach. A two-way table or tree diagram can help organise your thinking.
If a question asks 'how many' and the number of outcomes is likely to be large, the product rule is needed. Draw a box for each choice, write the number of options in it, and multiply them together.

Try the lesson

The start of the Number - Structure and calculation: apply systematic listing strategies, including use of the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be done is m × n ways) (Higher tier) lesson, made from checked teaching notes for this specification point.

Have a go first

A restaurant has 4 starters and 6 main courses. A customer chooses one starter and one main course. Using the product rule for counting, work out how many different two-course meals are possible.

Tip: press 1 to 4 to choose

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