Systematic Listing and Counting
Organise choices so that nothing is missed and nothing is counted twice. Begin with small lists, then use tables and trees to make your reasoning clear.
Foundation-first core. A clearly labelled Higher extension introduces the product rule and three-stage counting.
Start with a plan
A random list makes it easy to miss a choice or count one twice. A systematic list follows a clear order.
An outcome here means one complete valid choice, such as a drink with a snack.
Keep one choice fixed, list every permitted next choice, then move on. The structure helps you show that your list is complete.
Example Question:
How many different drink-and-snack choices come from two drinks and three snacks?
Example 1 lists all six before using a counting shortcut.
Exam Tip
Underline restrictions such as “no repeated digits” or “must be even”.
If asked to list, show the outcomes. A correct total alone does not supply the list.
Method — List once, count once
1. Decide what makes two outcomes different. Does order matter?
2. Choose an order for the first stage.
3. For each first choice, finish all permitted later choices.
4. Check conditions, then check for missing or repeated entries.
5. Count the complete outcomes. Use multiplication only when its stage counts fit the conditions.
Worked Examples
Example 1 — Group a list by its first choice
Choose one drink, tea or coffee, and one snack, cake, biscuit or fruit. Any snack can go with either drink.
Keep tea fixed and finish all its snacks before changing drinks.
| Drink | All snack choices |
|---|---|
| Tea | Tea + cake; tea + biscuit; tea + fruit |
| Coffee | Coffee + cake; coffee + biscuit; coffee + fruit |
There are 6 combinations. A combination here means one complete drink-and-snack choice.
The list is complete: each of the two drinks has all three snacks. No pair appears twice.
Example 2 — Use a two-way table
Choose a red, blue or green shirt and black or grey trousers. All pairings are allowed.
| Shirt / trousers | Black | Grey |
|---|---|---|
| Red | Red + black | Red + grey |
| Blue | Blue + black | Blue + grey |
| Green | Green + black | Green + grey |
Each inside cell is one outfit. The headings organise the choices; they are not extra outfits.
There are 6 outfits. Check each shirt row has both trouser colours.
This is a table of possibilities, not a table of frequencies.
Example 3 — Follow every complete tree path
Choose soup or salad as a starter, then pasta, rice or fish as a main. Every main is available after either starter.
Follow a path from Start, through one starter, to one main.
Soup + pasta; soup + rice; soup + fish.
Salad + pasta; salad + rice; salad + fish.
There are 6 complete paths. Each starter has all three branches, so no meal is missing.
Count complete paths, not all the lines in the diagram. We are counting choices; no probabilities are attached.
Example 4 — Read the repetition rule
Make two-digit numbers from 2, 4 and 7, without repeating a digit.
The first digit is the tens digit. The second is the units digit. In a number, order matters: 24 and 42 are different.
| First digit | Permitted numbers |
|---|---|
| 2 | 24, 27 |
| 4 | 42, 47 |
| 7 | 72, 74 |
All possibilities: 24, 27, 42, 47, 72, 74. There are 6.
Each first digit has the two different digits available for its second position.
If repetition is allowed, include 22, 44 and 77 too:
22, 24, 27; 42, 44, 47; 72, 74, 77. Now there are 9.
Only the repetition condition changed.
Example 5 — Apply an extra condition
Using 2, 4 and 7 without repetition, list all two-digit even numbers.
Even numbers end in 2 or 4. Check the units digit of every number in the previous complete list.
Starting with 2: 24 is allowed; 27 is odd.
Starting with 4: 42 is allowed; 47 is odd.
Starting with 7: 72 and 74 are both allowed.
Answer: 24, 42, 72, 74 — four numbers.
The first-digit groups contribute 1, 1 and 2 answers. Add these counts. Do not assume every group has the same size.
When order does not matter — both tiers
Choose two different people from A, B and C for a team with identical roles. AB and BA are the same team, so list each pair once: AB, AC, BC. There are 3 teams. If one person were captain and the other deputy, the roles would make order matter instead.
Quick practice — List the two-person teams from A, B, C and D, with no different roles.
The teams are AB, AC, AD, BC, BD, CD: 6 in total. Reversing a pair does not create another team.
Example 6 — Higher extension: explain the product rule
A badge has one of 3 shapes and one of 4 colours. Every colour is available for every shape.
For each shape, a systematic list would contain four badges.
4 + 4 + 4 = 12
This is 3 × 4 = 12 choices. Multiplication counts the equal-sized groups.
This shortcut is called the product rule for counting. For a fixed number of choices at each stage, multiply the numbers of permitted choices.
Check the conditions first. If the number of later choices varies, count the separate groups instead.
Example 7 — Higher extension: add a third stage
Choose one of 2 starters, then one of 3 mains, then one of 2 desserts. Every pairing is allowed.
The first two stages give 2 × 3 = 6 starter-and-main pairs.
Each of those six pairs has two dessert choices.
6 × 2 = 12 meals
So the total is 2 × 3 × 2 = 12. A complete three-stage tree would have 12 end paths.
Example 8 — Higher extension: a restricted code
A code has two letters followed by one digit. The letters are A, B or C, and the digit is 1, 2, 3 or 4. The two letters must be different.
The first letter has 3 choices. After it is used, the second letter has 2 choices. Any of the 4 digits can finish the code.
3 × 2 × 4 = 24 codes
Check with letter-pair groups: AB, AC, BA, BC, CA, CB.
Each of these six groups has codes ending 1, 2, 3 and 4. Six groups of four confirms all 24 exactly once.
Key ideas to remember
A table cell or complete tree path represents one outcome.
An organised list shows the possibilities. A product-rule calculation counts them efficiently.
Repetition allowed means a choice can be reused within an outcome. Do not assume this without reading the question.
Check completeness
Check every permitted first choice appears as a group.
Within each group, check every permitted next choice appears once.
Check each entry satisfies every restriction.
Finally, check different entries represent different outcomes. In digit numbers and codes, position matters.
Think Like a Mathematician
How can you know nothing is missing?
In the café list, any valid order must start with tea or coffee, so it belongs in one of the two groups.
Within that group, its snack must be cake, biscuit or fruit. All three are present.
Each pair has exactly one place in the list. That explains both completeness and the absence of duplicates.
Common mistakes to avoid
Listing randomly: group by the first choice so gaps become visible.
Counting twice: check whether two entries describe the same outcome.
Ignoring repetition: 22 is allowed only if the digit 2 can be reused.
Multiplying without checking restrictions: different-sized groups need separate counts.
Giving only a total: show all outcomes when the question asks for a list.
Counting tree lines: count complete paths. These diagrams need no probability labels.
Try it yourself
Show how your organisation includes each valid outcome exactly once. Questions 8–10 are the Higher product-rule extension.
1. Choose water or juice and an apple or a pear. List all possible pairs.
Fix water first: water + apple; water + pear.
Then juice: juice + apple; juice + pear.
4 pairs. Both drinks have both fruits exactly once.
2. Complete a table for red, blue or white hats with a small or large size. How many choices?
| Colour / size | Small | Large |
|---|---|---|
| Red | Red + small | Red + large |
| Blue | Blue + small | Blue + large |
| White | White + small | White + large |
6 choices. All three rows contain both sizes; each cell names one different choice.
3. Draw a tree for tea or juice, followed by cake, fruit or toast. List its end outcomes.
Tea + cake; tea + fruit; tea + toast.
Juice + cake; juice + fruit; juice + toast.
6 complete paths. Each drink has three snack branches, and each full path is listed once.
4. List all two-digit numbers made from 1 and 3 when repetition is allowed.
Fix the first digit at 1: 11, 13.
Fix it at 3: 31, 33.
11, 13, 31, 33 — four numbers. Each of the two starts has both endings.
5. List all two-digit numbers from 1, 3 and 5 without repeating a digit.
Starting 1: 13, 15. Starting 3: 31, 35. Starting 5: 51, 53.
6 numbers. Each first digit has both different digits as endings. No digit repeats within a number.
6. From 1, 3 and 5, list two-digit numbers greater than 30 without repetition.
Start with the complete groups: 13, 15; 31, 35; 51, 53.
Remove the group starting with 1, as both are below 30.
Answer: 31, 35, 51, 53. The two remaining starts each have both permitted endings.
7. A café offers tea or coffee and cake, fruit or toast, but toast is unavailable with coffee. List all valid choices.
Tea: tea + cake; tea + fruit; tea + toast.
Coffee: coffee + cake; coffee + fruit.
5 choices, from groups of 3 and 2. The forbidden coffee + toast is excluded; every other pair is present.
8. Higher: a bag comes in 4 colours and 3 sizes. Every colour has every size. How many choices?
Each of the four colours gives three distinct size choices.
4 × 3 = 12 choices. Four complete rows of three would list every choice once.
9. Higher: choose one of 2 breads, 3 fillings and 2 drinks. All choices are available together. How many lunches?
2 × 3 = 6 bread-and-filling pairs.
Each pair has 2 drinks: 6 × 2 = 12 lunches.
A full list has six different first-two-stage pairs, each appearing once with each drink.
10. Higher: a code uses two different letters from A, B and C, then digit 1 or 2. How many codes?
Letter pairs: AB, AC, BA, BC, CA, CB.
The full list is AB1, AB2; AC1, AC2; BA1, BA2; BC1, BC2; CA1, CA2; CB1, CB2.
3 × 2 × 2 = 12 codes. Each valid letter pair has both digits once; repeated letters are excluded.
Frequently Asked Questions
What does systematic mean?
It means following a planned order, such as completing one first-choice group before moving to the next.
Are 24 and 42 different outcomes?
Yes, when forming numbers: the digits occupy different positions and the values differ.
Must I use a tree?
No. A list or table may be clearer. Use the representation requested by the question, if one is specified.
Can I always multiply the original numbers of choices?
No. Restrictions may remove choices. Check how many remain at each stage or count separate groups.
Is this probability?
This lesson counts valid outcomes. It does not calculate how likely they are.
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