The idea, step by step
Count choices stage by stage
If there are m choices for a first step and n choices for a second step for every first choice, there are mn complete choices. A tree explains why: each first branch splits into n second branches.
Identify exactly what is being made. An ordered code can start with zero, but a three-digit whole number cannot. Repetition may be allowed for a code but not for choosing different people.
Restrictions change later choices
When objects cannot be reused, the available choices decrease. A three-letter code using different letters from A,B,C,D has 4×3×2 possibilities. If repeats are allowed, it has 4×4×4.
If a restriction divides a problem into cases, count each non-overlapping case and then add. Do not multiply counts for choices that cannot be combined, or add overlapping cases.
A common trap
Using 4×3×2 for the three-digit-number example with digits 0–3.
Instead: The first digit cannot be zero, so it has only three choices.
See the thinking
Menu choices
A lunch has one of 3 mains, 2 drinks and 4 fruit choices. How many lunches?
- Choose a main in 3 ways.
- Each combines with 2 drinks and then 4 fruits.
- 3×2×4=24.
A zero cannot lead
How many three-digit numbers use different digits chosen from 0,1,2,3?
- The first digit has 3 choices:1,2,3.
- The next has 3 unused choices, including zero.
- The final digit has 2 choices.
Check your understanding
How many two-letter codes use A,B,C with repeats allowed?
Show answer
9.
There are three choices for each of two positions:3×3.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- There are 4 shirts and 3 trousers. How many outfits choosing one of each?2 marks
- A four-digit PIN allows 0–9 in each position and permits repeats. How many PINs?2 marks
- How many three-letter codes use different letters from A,B,C,D,E?2 marks
- Five runners fill first and second place with no ties. How many ordered results?2 marks
- How many two-digit numbers use distinct digits from 0,2,4,6?3 marks
- A student says 3 mains and 2 drinks make 5 lunches. Explain and correct.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- There are 4 shirts and 3 trousers. How many outfits choosing one of each?2 marks
4×3=12.
12. - A four-digit PIN allows 0–9 in each position and permits repeats. How many PINs?2 marks
10^4=10,000; a PIN may begin with zero.
10,000. - How many three-letter codes use different letters from A,B,C,D,E?2 marks
5×4×3=60.
60. - Five runners fill first and second place with no ties. How many ordered results?2 marks
5 choices for first and 4 for second.
20. - How many two-digit numbers use distinct digits from 0,2,4,6?3 marks
3 choices for the tens digit, then 3 remaining choices.
9. - A student says 3 mains and 2 drinks make 5 lunches. Explain and correct.3 marks
Every main combines with each drink.
Multiply 3×2 rather than adding separate categories.
There are 6 lunches.
Curriculum reference & lesson scope
Australian Curriculum v 9 / SA Curriculum
Compound events; systematic counting and the multiplication principle; factorials and simple arrangements; conditional language and restricted sample spaces; conditional probabilities from tables/trees; replacement and dependence; conditional versus unconditional probabilities; designing and interpreting simulations.
Official curriculum source ↗