Multiplication principle | Year 11

01 / LEARN

Understand the idea.

Count complete outcomes, not individual options

A complete outcome includes a choice at every stage. For a meal, choosing a main alone is unfinished: a side and drink are still needed. Count the possibilities for each position, then multiply to count all complete combinations.

Imagine a tree with one branch for each main, then branches for sides and drinks. Multiplication counts the final branches without drawing them all, provided every partial outcome at a given stage has the stated number of continuations.

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

Update choices when a restriction changes the next stage

Restrictions change the number available at later stages. Without repetition, each used object removes one option. A three-digit number cannot begin with zero, although zero may appear in a later position.

If branch sizes differ, split the outcomes into disjoint cases, multiply within each case and add the case totals. The multiplication principle does not require independent choices; it requires a correct count of legal continuations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A complete meal

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

Worked solution

1
Choose a main.
$n_{1} = 4$
Each main starts a different complete meal.
2
Extend each choice with a side.
$4 \times 3 = 12$
All three sides are available with every main.
3
Include the drink.
$N = 12 \times 2 = 24$
Each main-and-side pair has two drink choices, giving 24 meals.

Answer

24 meals.

EXAMPLE 02

A restricted number

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

Worked solution

1
Count legal first digits.
$n_{1} = 4$
Only 1, 2, 3 and 4 can occupy the hundreds position.
2
Count the remaining positions.
$\begin{aligned} n_{2} & = 5 - 1 = 4 \\ n_{3} & = 5 - 2 = 3 \end{aligned}$
Zero is now allowed, but neither previously used digit can be repeated.
3
Multiply the position counts.
$N = 4 \times 4 \times 3 = 48$
Every legal three-digit number has exactly one sequence of these choices.

Answer

48 numbers.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

12.

Each of the three shirts can pair with each of four trousers. Therefore there are $3 \times 4 = 12$ complete outfits.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Count complete outcomes, not individual options

Update choices when a restriction changes the next stage

See the method in action.

A complete meal

Choose a main.

Extend each choice with a side.

Include the drink.

A restricted number

Count legal first digits.

Count the remaining positions.

Multiply the position counts.

Try it yourself.

Check your understanding

Keep practising