Systematic listing and the product rule
Edexcel 1MA1 expects students to count outcomes without missing any and without double-counting. The skill is tested at Foundation as ordered listing and at Higher as the product rule (sometimes called the multiplication principle).
Systematic listing — Foundation focus
A list is systematic when each entry follows a clear rule so the reader can see no duplicates exist.
Example: list all 2-digit numbers using digits 3, 5, 7 with no repeats.
- Fix first digit 3: 35, 37 (2 numbers)
- Fix first digit 5: 53, 57 (2 numbers)
- Fix first digit 7: 73, 75 (2 numbers)
Total: 6 outcomes — and the listing convention itself shows you have not missed any.
The product rule — Higher focus
If event A has m outcomes and event B has n independent outcomes, the combined event A then B has m × n outcomes.
Example: a 4-digit PIN where each digit is 0–9 has 10 × 10 × 10 × 10 = 10,000 PINs.
Example with restrictions: a 4-digit PIN with no repeated digits has 10 × 9 × 8 × 7 = 5,040 PINs.
Example with extra constraint (Edexcel Higher style): A car number plate has 2 letters then 3 digits. The first letter cannot be I, O, Q. The first digit cannot be 0. How many plates exist?
Plates = 23 × 26 × 9 × 10 × 10 = 538,200.
When listing beats the product rule
Listing is required when outcomes have dependencies (e.g. "the second digit must be greater than the first"). In those cases, draw a table or list rather than blindly multiplying.
Common Edexcel mark-scheme phrasing
- "Listing 6 outcomes systematically" earns M1.
- "All 6 correct" earns A1.
- For product rule, the M1 typically goes to a correct expression like "4 × 3 × 2" before evaluation. The A1 is for the final value.
⚠Common mistakes— Common errors
- Multiplying when one event depends on another (use listing instead).
- Forgetting that a 4-digit PIN starting with 0 is still valid (do not subtract).
- Treating "with replacement" the same as "without replacement" — read the question.
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