Multiplication principle | Year 11

01 / LEARN

The idea, step by step

Count complete outcomes, not individual options

An outcome may contain several choices: a meal can have a main, a side and a drink. If every main can be paired with every side and every drink, multiply the numbers of options. The product counts complete meals, not the total number of menu entries. A tree or small table can make the combinations visible.

N = n_{1} n_{2} \dots n_{k}

Update choices when a restriction changes the next stage

Independence of choices is not required for the multiplication principle. What matters is knowing how many continuations each partial choice has. For a code without repeated digits, the next stage has one fewer option. If different branches have different numbers of continuations, count the separate cases and add them; do not multiply one arbitrary branch size.

A common trap

Adding the numbers of options for stages of one outcome.

Instead: Multiply for successive choices; add only disjoint alternative cases.

02 / WORKED EXAMPLES

See the thinking

Example 1

A complete meal

A cafe offers 4 mains, 3 sides and 2 drinks. Every combination is available. How many meals are possible?

Choose a main in 4 ways.
For each main, choose a side in 3 ways and a drink in 2 ways.
$4 \times 3 \times 2 = 24$ .

24 meals.

Example 2

A restricted number

How many three-digit numbers can be made from 0, 1, 2, 3, 4 without repetition?

The first digit cannot be zero: 4 choices.
After choosing it, 4 digits remain, then 3.
$4 \times 4 \times 3 = 48$ .

48 numbers.

Check your understanding

There are 3 shirts and 4 trousers. How many outfits?

Show answer

12.

Each shirt can be paired with all 4 trousers: $3 \times 4 = 12$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

There are 5 sandwich fillings and 3 bread types. Count the sandwiches.
2 marks
A four-digit PIN may use digits 0–9 repeatedly. Count the PINs.
2 marks
A four-digit PIN has no repeated digits. Count it.
2 marks
Choose a captain and then a different vice-captain from 8 players. Count the choices.
2 marks
A password is either two letters A–D followed by a digit 0–9, or one letter A–D followed by two digits. Repetition is allowed. Count all passwords.
3 marks
A student claims three tasks with 4, 2 and 3 choices have $4 + 2 + 3 = 9$ complete outcomes. Explain and correct the error.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

There are 5 sandwich fillings and 3 bread types. Count the sandwiches.
2 marks
$5 \times 3 = 15$ .
15.
A four-digit PIN may use digits 0–9 repeatedly. Count the PINs.
2 marks
A PIN may start with zero.
$10^{4} = 10000$ .
10,000.
A four-digit PIN has no repeated digits. Count it.
2 marks
$10 \times 9 \times 8 \times 7 = 5040$ .
5,040.
Choose a captain and then a different vice-captain from 8 players. Count the choices.
2 marks
The roles differ.
$8 \times 7 = 56$ .
56.
A password is either two letters A–D followed by a digit 0–9, or one letter A–D followed by two digits. Repetition is allowed. Count all passwords.
3 marks
The formats cannot overlap.
$4^{2} \times 10 + 4 \times 10^{2} = 160 + 400 = 560$ .
560.
A student claims three tasks with 4, 2 and 3 choices have $4 + 2 + 3 = 9$ complete outcomes. Explain and correct the error.
3 marks
Adding counts options separately rather than complete triples.
$4 \times 2 \times 3 = 24$ .
24 outcomes; each choice in one stage can combine with every choice in the next.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.

Official curriculum source ↗

Back to Counting and introductory statistics