P6: Systematic Listing
Enumerate sets and combinations of sets; Venn diagrams; systematic listing strategies
📋 Key Concepts
Systematic Listing: A method of listing all possible outcomes in an organised way to ensure none are missed or repeated.
Key Terms
| Term | Symbol | Meaning |
|---|---|---|
| Union | A ∪ B | Elements in A OR B (or both) |
| Intersection | A ∩ B | Elements in A AND B |
| Complement | A' | Elements NOT in A |
| Universal Set | ξ or U | All possible elements |
📝 Systematic Listing Strategies
Method: Fix one element and vary the other systematically. This ensures you don't miss any combinations.
Example 1
A meal consists of one sandwich and one drink. Sandwiches: Ham (H), Cheese (C), Tuna (T). Drinks: Water (W), Juice (J). List all possible meals.
Systematic listing:
Fix Ham: H-W, H-J
Fix Cheese: C-W, C-J
Fix Tuna: T-W, T-J
6 possible meals
Using product rule: 3 × 2 = 6 ✓
Example 2
List all 2-digit numbers that can be made using digits 2, 5, 8 if repetition is allowed.
Systematic listing:
Fix first digit as 2: 22, 25, 28
Fix first digit as 5: 52, 55, 58
Fix first digit as 8: 82, 85, 88
9 numbers: 22, 25, 28, 52, 55, 58, 82, 85, 88
📝 Product Rule for Counting
Rule: If there are m ways to do one thing and n ways to do another, there are m × n ways to do both.
Total combinations = m × n × p × ...
Example 3
A password uses 3 letters. How many possible passwords are there?
Solution:
First letter: 26 choices
Second letter: 26 choices
Third letter: 26 choices
Total = 26 × 26 × 26 = 17,576 passwords
📝 Venn Diagrams
Venn Diagrams: Visual representation of sets showing overlap (intersection) and total coverage (union).
Example 4
In a class of 30: 18 study Maths, 14 study English, 7 study both. Draw a Venn diagram.
Solution:
┌─────────┬─────────┐
│ Maths │ English │
│ 11 │ 7 │ 11 = Maths only (18 - 7)
│ │ │ 7 = Both
├─────────┴─────────┤ 7 = English only (14 - 7)
│ Both: 7 │ 5 = Neither (30 - 11 - 7 - 7)
│ English: 7 │
│ Neither: 5 │
└───────────────────┘
Check: 11 + 7 + 7 + 5 = 30 ✓
📝 Set Notation
Notation: Use proper set notation to describe regions in Venn diagrams.
Example 5
Given sets A and B, describe each region:
a) A ∩ B (A and B both)
b) A ∪ B (A or B or both)
c) A' (not in A)
d) A ∩ B' (in A but not B)
Descriptions:
a) Elements that are in BOTH A and B (middle overlap)
b) Elements in A, B, or both (both circles)
c) Elements NOT in A (outside circle A)
d) Elements in A ONLY (left crescent, not in B)
📝 Finding Probabilities from Sets
Example 6
In a group of 50 people: 30 like pizza, 25 like burgers, 15 like both. Find:
a) P(likes pizza only)
b) P(likes neither)
c) P(likes pizza or burgers)
Solution:
Pizza only = 30 - 15 = 15 people
Burgers only = 25 - 15 = 10 people
Both = 15 people
Neither = 50 - 15 - 10 - 15 = 10 people
a) P(pizza only) = 15/50 = 0.3
b) P(neither) = 10/50 = 0.2
c) P(pizza or burgers) = (15 + 10 + 15)/50 = 40/50 = 0.8
🧮 Worked Examples — Exam Style
Counting with sets: product rule for stages; Venn diagrams for overlapping groups (add only, subtract the intersection); set notation n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
In a class of 30: 18 study Spanish, 15 study French, 6 study both. How many study neither?
Spanish or French: 18 + 15 − 6 = 27 → neither: 30 − 27 = 3
M1 for the union formula, A1 for 3. Draw the Venn: 12 + 6 + 9 in the three regions, 3 outside.
How many ways can 3 different prizes be awarded to 10 people (one each)?
10 × 9 × 8 = 720 ways
M1 for the product, A1 for 720. Prizes are distinct → order matters → descending product.
Exam wording to expect: "how many possible …", Venn diagram shading/completion, set notation.
❓ Practice Questions
Q1: A lunch has 3 sandwich options and 4 drink options. How many possible lunches?
Q2: List all outcomes when spinning two 2-sided spinners showing A and B.
Q3: In a survey of 40: 22 have cats, 18 have dogs, 8 have both. How many have neither?
Q4: In a Venn diagram with sets A and B, shade the region A' ∩ B.
Q5: A 4-digit PIN uses digits 0-9. How many possible PINs?
Q6: A 4-digit PIN cannot start with 0. How many possible PINs?
Q7: Universal set 1–20. A = multiples of 3, B = even numbers. Find n(A ∩ B) and n(A ∪ B).
✅ Answers
- 3 × 4 = 12 possible lunches
- AA, AB, BA, BB (4 outcomes)
- Cats only = 22-8 = 14, Dogs only = 18-8 = 10, Both = 8, Neither = 40-14-10-8 = 8
- The region in B but NOT in A (right crescent)
- 10 × 10 × 10 × 10 = 10,000 possible PINs
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 3 × 4 = 12
Q2. AA, AB, BA, BB — 4 outcomes
Q3. Only cats 14, only dogs 10, both 8 → neither = 40 − 32 = 8
Q4. In B but not A — B minus the overlap (right crescent)
Q5. 10⁴ = 10,000
Q6. 9 × 10 × 10 × 10 = 9000
Q7. A ∩ B = multiples of 6 ≤ 20: {6, 12, 18} → 3; n(A) = 6, n(B) = 10 → union = 6 + 10 − 3 = 13
📌 Key Facts to Memorise
- Product rule: multiply choices per stage (independent stages)
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B) — subtract the overlap once
- Venn regions: only ∩ counted twice — fill the INTERSECTION first, then subtract outwards
- Set notation: ∪ or · ∩ and · A′ not-A
- Restricted digits (no leading zero): multiply the reduced first-stage choice
🗺️ In the Exam
Counting with sets: 3–5 marks on Papers 1/2 — PINs, menus, Venn completions. The Venn-fill order (centre first) prevents every arithmetic slip.
Links to: listing (N5), mutual exclusivity (P4), probability from sets (P3).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always list combinations systematically to avoid missing any
- Use the product rule to check your listing (multiply choices)
- In Venn diagrams, work from the inside out (intersection first)
- Check the total in a Venn diagram equals the universal set
- Shade carefully when asked to identify regions in Venn diagrams
🧠 Problem-Solving Strategies
Problem-Solving
For systematic listing and Venn diagram problems: (1) Fix one variable and vary the other to list systematically, (2) Use the product rule to check your total: choices × choices = total combinations, (3) For Venn diagrams, always fill in the intersection FIRST, then work outwards, (4) Check that all regions in a Venn diagram sum to the universal set total.
Multi-Step Problem
In a group of 40 people: 22 have a cat, 18 have a dog, 8 have both. (a) Draw a Venn diagram. (b) Find P(has a cat only). (c) Find P(has neither). (d) Find P(has a cat given they have a dog).
Solution: (a) Both = 8. Cat only = 22 − 8 = 14. Dog only = 18 − 8 = 10. Neither = 40 − 14 − 8 − 10 = 8. (b) P(cat only) = 14/40 = 7/20. (c) P(neither) = 8/40 = 1/5. (d) P(cat | dog) = 8/18 = 4/9.
⚠️ Common Errors
Watch Out!
1. Wrong: In a Venn diagram, putting 22 in the cat circle and 18 in the dog circle without subtracting the overlap Correct: Cat only = 22 − 8 = 14. The 8 who have both go in the intersection, not counted twice
2. Wrong: Confusing A ∩ B (intersection/AND) with A ∪ B (union/OR) Correct: ∩ means elements in BOTH (the overlap), ∪ means elements in EITHER (all of both circles)
3. Wrong: Missing combinations when listing by not using a systematic method Correct: Always fix one element and vary the other — check your count with the product rule
✍️ 6-Mark Exam Question
Extended Answer
6 marks: In a year group of 100 students: 54 study French, 42 study German, 18 study both. (a) Draw a Venn diagram showing these numbers. (b) Find the probability that a randomly chosen student studies exactly one language. (c) A student is chosen from those who study German. Find the probability they also study French.
(a) French only = 54 − 18 = 36. German only = 42 − 18 = 24. Both = 18. Neither = 100 − 36 − 24 − 18 = 22.
(b) Exactly one language = 36 + 24 = 60. P(exactly one) = 60/100 = 3/5 = 0.6.
(c) P(French | German) = P(both)/P(German) = 18/42 = 3/7 ≈ 0.429.
Mark scheme: M1 for subtracting overlap, A1 for correct Venn diagram, M1 for 36+24=60, A1 for 3/5, M1 for conditional probability, A1 for 3/7
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A menu offers 3 starters (S1, S2, S3), 4 mains (M1, M2, M3, M4), and 2 desserts (D1, D2). A meal deal lets you pick one of each.
(a) How many different meal combinations are possible?
(b) If you randomly choose a meal, what is the probability it includes M3?
(c) The restaurant removes one starter. Does this reduce the total combinations by more or less than removing one main? Justify your answer.
Answers: (a) 3 × 4 × 2 = 24 combinations. (b) With M3: 3 × 1 × 2 = 6 meals. P(includes M3) = 6/24 = 1/4. (c) Removing a starter: 2 × 4 × 2 = 16 (loss of 8). Removing a main: 3 × 3 × 2 = 18 (loss of 6). Removing a starter reduces more because it had more options (4 mains to pair with).
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