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
❓ 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?
✅ 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
🎯 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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