Understand the idea.
Count choices stage by stage
The multiplication principle counts complete choices stage by stage. If every first choice has the same number of available second choices, multiply these counts. A tree shows why: every first-stage branch leads to that many completed outcomes.
Define exactly what is being selected before counting. A code may begin with zero, whereas a multi-digit whole number cannot. Distinct roles make order meaningful; selecting a group may not. Restrictions determine the choices available at each stage.
Restrictions change later choices
Without reuse, count only the objects still available. Three different letters chosen in order from four give possibilities. If repeats are allowed, the availability resets and the count becomes .
When branch counts differ, split the problem into disjoint cases and add their totals. Multiplication combines successive choices in one outcome; addition combines alternative non-overlapping cases. Check that no valid outcome is omitted or counted in two cases.
See the method in action.
Menu choices
A lunch has one of 3 mains, 2 drinks and 4 fruit choices. How many lunches?
Worked solution
- 1
Count main-and-drink combinations.
Every main can be paired with either drink.
- 2
Include the fruit choice.
Each of the six combinations can be completed with any of four fruits.
- 3
State what was counted.
A lunch contains exactly one choice from each of the three categories.
24.
A zero cannot lead
How many three-digit numbers use different digits chosen from 0,1,2,3?
Worked solution
- 1
Choose a valid hundreds digit.
Zero cannot lead a three-digit whole number.
- 2
Choose the tens digit.
One of the four digits has been used; zero is now allowed.
- 3
Choose the units digit and multiply.
Only two unused digits remain for the final position, and each valid prefix has that many completions.
3×3×2=18.
Try it yourself.
Check your understanding
How many two-letter codes use A,B,C with repeats allowed?
Show answer & reasoning
9.
Each of two ordered positions has three choices because repeats are allowed: codes.
Keep practising
6 worksheet questions · 14 marks
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
.
12. - A four-digit PIN allows 0–9 in each position and permits repeats. How many PINs?2 marks
; a PIN may begin with zero.
10,000. - How many three-letter codes use different letters from A,B,C,D,E?2 marks
.
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 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 ↗