Sheet № 242 · Foundation + Higher · AQA · Edexcel · OCR
Systematic Listing and Product Rule –
Systematic listing and the product rule for counting are tested on GCSE Maths papers across AQA, Edexcel and OCR at Foundation and Higher tier. These skills ensure you can list every possible outcome of a combined event without missing any or repeating them, and quickly calculate the total number of outcomes without writing them all out.
§Key definitions
Systematic listing
means writing out all possible outcomes in an organised way so that none are missed and none are repeated. You fix one element and vary the others methodically.
Question:
A cafe offers 3 sandwiches (ham, cheese, tuna) and 4 drinks (tea, coffee, juice, water). How many different lunch combinations of one sandwich and one drink are possible?
Answer:
12 different lunch combinations.
(a)
Product rule: 3 × 4 × 2 = 24 outfits.
(b)
Fix black skirt and trainers. Only the top varies: Black-Red-Trainers, Black-White-Trainers, Black-Blue-Trainers, Black-Green-Trainers.
§Formulas to memorise
Total outcomes = m × n
Total outcomes = n₁ × n₂ × n₃ × ...
Systematic listing: means writing out all possible outcomes in an organised way so that none are missed and none are repeated. You fix one element and vary the others methodically.
Identify the categories — (e.g. starter, main, dessert or digit 1, digit 2).
Fix the first category — at its first option.
List all combinations — for the remaining categories.
Move to the next option — for the first category and repeat.
Continue — until all options for the first category are exhausted.
Count — the total and check it matches the product rule.
Worked example
A cafe offers 3 sandwiches (ham, cheese, tuna) and 4 drinks (tea, coffee, juice, water). How many different lunch combinations of one sandwich and one drink are possible?
Working:
⚠ Common mistakes
- ✗Missing combinations when listing. Without a systematic approach, students skip outcomes. Always fix one category and cycle through the others.
- ✗Forgetting "without replacement" restrictions. If digits or items cannot be repeated, reduce the number of choices at each stage.
- ✗Using the product rule when order does not matter. If the question asks for combinations (where order is irrelevant), the product rule overcounts — you may need to divide by the number of arrangements. At GCSE, most questions treat order as mattering.
✦ Exam tips
- →If the question says "list all possibilities", you must write every outcome — the product rule alone will not earn full marks.
- →If it says "how many", use the product rule for speed and show your multiplication.
- →When digits cannot repeat, think of it as choosing for each position in turn: the number of choices decreases by 1 at each step.
- →Check your list total matches the product rule to confirm you have not missed any.
- →For related probability, see sample space diagrams and combined events probability.