Systematic Listing
Systematic Listing
Systematic listing is the process of recording all possible outcomes of a situation in a structured, organised way. The two goals are completeness — no outcome is missed — and non-repetition — no outcome is listed twice. Achieving both requires a consistent ordering strategy rather than listing outcomes as they come to mind.
Systematic listing underpins probability, combinatorics and problem-solving throughout GCSE Mathematics. The three principal tools are: ordered lists, two-way tables (sample space diagrams), and tree diagrams. Which tool to use depends on the number of events and the size of the outcome space.
Two-Way Tables (Sample Space Diagrams)
When two independent events each have a small number of outcomes, a two-way table shows every combination in a grid. The rows represent outcomes of one event; the columns represent outcomes of the other. Every cell holds exactly one outcome, making completeness automatic.
The table below shows all outcomes when a fair coin is flipped and a standard die is rolled. There are \( 2 \times 6 = 12 \) equally likely outcomes.
| Coin Die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| H | (H,1) | (H,2) | (H,3) | (H,4) | (H,5) | (H,6) |
| T | (T,1) | (T,2) | (T,3) | (T,4) | (T,5) | (T,6) |
Reading off the table directly answers probability questions. For example, outcomes giving Heads and an even number: (H,2), (H,4), (H,6) — 3 outcomes out of 12.
Worked Examples
List all arrangements of the letters P, Q, R.
Fix the first letter and list the remaining two in both orders. Work through each first-letter option in turn:
P first: PQR, PRQ Q first: QPR, QRP R first: RPQ, RQP
Total: 6 arrangements. The fix-first approach guarantees completeness.
List all ways to choose 2 colours from {Red, Blue, Green}.
In combinations, order does not matter (Red+Blue is the same as Blue+Red). Fix the first item alphabetically and pair with each item that comes after it:
Blue+Green, Blue+Red, Green+Red
Total: 3 combinations.
Two spinners: A shows 1, 2, 3 and B shows 1, 2. Record all sums.
| A B | B=1 | B=2 |
|---|---|---|
| A=1 | 2 | 3 |
| A=2 | 3 | 4 |
| A=3 | 4 | 5 |
All \( 3 \times 2 = 6 \) outcomes visible. A sum of 3 occurs twice, so \( P(\text{sum} = 3) = \frac{2}{6} = \frac{1}{3} \).
Form 2-digit numbers from {3, 5, 7, 9} where tens digit is greater than units digit.
Tens = 5: 53. Tens = 7: 73, 75. Tens = 9: 93, 95, 97.
(Tens = 3 produces no valid pair since no digit in the set is less than 3.)
Total: 6 numbers.
Key Takeaways
- Always work in a fixed order — alphabetical, numerical or positional — to ensure completeness and avoid repetition.
- For arrangements (order matters), fix the first element and vary the rest; for combinations (order does not matter), only move forward through the remaining options.
- Two-way tables are ideal for two independent events — every cell is one outcome.
- Apply constraints from the start of the listing method rather than filtering afterwards.
- Verify the total count by counting table cells or tree-diagram endpoints.