Why This Topic Matters
Counting appears in many forms. Sometimes it looks like a simple list. Sometimes it looks like a team-selection question. Sometimes it hides inside another chapter such as probability.
At Higher Level, this topic goes one step further. You still need the same basic counting ideas, but you also use them in more advanced algebra, especially binomial expansion.
The hardest part is often not the arithmetic. The hardest part is deciding what type of counting problem you are looking at. That is why these notes keep returning to simple questions like: “Am I arranging things?” “Am I just choosing a group?” “Does changing the order change the answer?”
Read the question like a real situation. What is actually happening?
Decide whether you need listing, stage multiplication, permutations, or combinations.
Repeated letters, odd endings, or fixed groups must be handled carefully from the start.
Learning Objectives
- Write sample spaces clearly when needed.
- Use multiplication to count multi-stage choices.
- Recognise when order matters.
- Recognise when order does not matter.
- Use Pascal's Triangle and binomial coefficients for Higher Level work.
Formula Toolkit
Use when a task happens in stages.
Use when order matters.
Use when order does not matter.
These numbers become the coefficients in the expansion.
Interactive Simulators
These tools help the topic make sense. Use them to test patterns and check whether your thinking matches the method.
Counting Builder
Type the number of choices at each stage. The tool multiplies them and shows the total outcomes.
Permutation vs Combination Explorer
Use this to compare the two ideas. The numbers may look similar, but the meaning is different.
Pascal Row and Binomial Coefficients
Pick a power and the row appears automatically. This helps you connect combinations to algebra.
Which Tool Should I Use?
When you feel stuck, come back to this guide.
- List outcomes: sample space, table, or tree diagram
- Multiply stages: Fundamental Principle of Counting
- Arrange positions: permutation
- Choose a group: combination
- Expand a bracket: binomial coefficients
Before We Start: Permutation or Combination?
Students often mix these up, so let us settle the difference clearly. A permutation is about arrangement. A combination is about selection.
Imagine three students: Ahmed, Sara, and Lina. If we are choosing a 2-person team, Ahmed and Sara is the same team as Sara and Ahmed. That means the order does not matter.
But if we are choosing first place and second place, Ahmed first and Sara second is not the same as Sara first and Ahmed second. That means order does matter.
| Question type | What matters? | Method |
|---|---|---|
| Choose a 3-person committee | Only who is in the group | Combination |
| Choose gold, silver, and bronze | Who is in each position | Permutation |
| Choose 4 toppings for a pizza | The set of toppings | Combination |
| Make a 4-letter code | The order of the letters | Permutation or stage multiplication |
1. Listing Outcomes Systematically
Sometimes the best thing to do is simply list every possible result in an organised way. This is called writing the sample space.
The key word is systematically. That means you follow a pattern so that nothing is missed and nothing is repeated.
Flip one coin.
The outcomes are H and T.
Total outcomes = 2
Flip one coin and roll one die.
Each coin result pairs with 6 die results.
Total outcomes = 12
2. The Fundamental Principle of Counting
This principle is simpler than it sounds. If a task happens in stages, and each stage has a number of choices, then the total number of full outcomes is found by multiplying.
3 shirts and 4 pairs of trousers.
Total outfits = 3 × 4 = 12
3 starters, 5 mains, 2 desserts.
Total meals = 3 × 5 × 2 = 30
3. Arrangements (Permutations)
A permutation is used when the order matters. This means that swapping positions gives a different answer.
8 runners compete for gold, silver, and bronze.
8P3 = 8 × 7 × 6 = 336
5 different books on a shelf.
5! = 120
4. Selections (Combinations)
A combination is used when you are choosing a group and the order does not matter.
Choose 4 people from 10.
10C4 = 210
Choose 3 boys from 5 and 2 girls from 6.
5C3 × 6C2 = 150
5. Binomial Coefficients (Higher Level)
At Higher Level, counting connects to algebra. The same numbers that count combinations also appear in Pascal's Triangle and in expansions such as (x + y)n.
Row 3 is 1, 3, 3, 1.
These numbers match 3C0, 3C1, 3C2, and 3C3.
(x + y)3 = x3 + 3x2y + 3xy2 + y3
- Ask which term gives x3.
- That means the power on 2x must be 3.
- So n - r = 3, therefore r = 2.
- Then calculate the coefficient carefully.
- Coefficient = 720
Exam-Style Questions
EasyQ1: A restaurant has 4 starters, 6 mains, and 3 desserts. How many different 3-course meals are possible?
MediumQ2: How many 5-digit odd numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7 if no digit repeats?
CoreQ3: A committee of 4 is chosen from 5 teachers and 7 students. In how many ways can the committee contain exactly 2 teachers and 2 students?
HigherQ4: Find the coefficient of x2 in the expansion of (x + 2)4.
A1: 4 × 6 × 3 = 72
A2: Final digit must be odd, so start there. Then arrange the other positions from the remaining digits. Total = 1440
A3: Choose 2 teachers and 2 students: 5C2 × 7C2 = 210
A4: The coefficient of x2 in (x + 2)4 is 24.
One-Page Summary
- Use listing when the question wants the sample space.
- Use multiplication for stage-by-stage choices.
- Use permutations when order matters.
- Use combinations when order does not matter.
- At Higher Level, binomial coefficients link counting and algebra.
Do not start with “Which formula do I remember?” Start with “What is happening in this question?” That small change makes counting much easier.