YEAR 10 · PROBABILITY · 18 MIN

Factorials and simple arrangements

Use factorials for ordered arrangements and identify when order creates overcounting.

In this lesson

Evaluate factorial expressions
Count ordered arrangements
Explain simple overcounting

Before you start

Fractions, sample spaces and probability

01 / LEARN

Understand the idea.

Factorial represents decreasing choices

For $n$ different objects placed in a row, the choices decrease from $n$ to one. Their product is factorial, $n!$ . The last position has one choice, and the convention $0! = 1$ also makes cancellation formulas work when no positions remain.

Use only as many factors as there are positions to fill. A captain and deputy are distinct roles, so choosing them from six people gives $6 \times 5$ , not $6!$ . The unselected people are not being arranged.

n! = n (n - 1) \dots 2 \cdot 1

Identical objects and unordered groups

Factorial counts distinguishable objects in ordered positions. Identical letters require care: swapping the two A copies in AAB creates no visibly new arrangement. Listing AAB, ABA and BAA exposes the overcount from treating the copies as different.

For an unordered pair, exchanging the two people does not change the group. Count the ordered selections, then divide by two because each group appears in exactly two orders. Identify that repeated counting before deciding what to divide by.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Arrange different books

Five different books are placed in a row. How many orders?

Worked solution

1
Count the first two positions.
$5 \times 4$
After selecting one book, only four of the five different books remain.
2
Include the remaining positions.
$5! = 5 \times 4 \times 3 \times 2 \times 1$
No book is reused, so the available choices decrease by one each time.
3
Evaluate the product.
$5! = 120$
Every completed row is counted once by its successive book choices.

Answer

120.

EXAMPLE 02

Roles versus a group

From 5 people, compare choosing a captain and deputy with choosing an unordered pair.

Worked solution

1
Choose the two distinct roles.
$5 \times 4 = 20$
There are five captain choices and four different people available as deputy.
2
Identify the repeated counting for a group.
$(A, B) and (B, A)$
These are different role assignments but the same unordered pair.
3
Divide by the number of orders per pair.
$\frac{20}{2} = 10$
Each two-person group occurred exactly twice among the twenty assignments.

Answer

20 role assignments;10 pairs.

03 / YOUR TURN

Try it yourself.

Check your understanding

Evaluate 6!/4!.

Show answer & reasoning

30.

Expand only until the denominator cancels: $\frac{6!}{4!} = \frac{6 \times 5 \times 4!}{4!} = 6 \times 5 = 30$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $4!$ .
2 marks
Evaluate $\frac{7!}{5!}$ .
2 marks
Six different students line up. How many orders?
2 marks
From 6 students choose a captain and deputy. How many choices?
2 marks
How many unordered two-person groups can be chosen from 6 students?
3 marks
List the distinct arrangements of AAB and explain why there are not 6.
3 marks

Read the worked solutions

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

Evaluate $4!$ .
2 marks
$4 \times 3 \times 2 \times 1 = 24$ .
24.
Evaluate $\frac{7!}{5!}$ .
2 marks
Cancel $5!$ , leaving $7 \times 6$ .
42.
Six different students line up. How many orders?
2 marks
$6! = 720$ .
720.
From 6 students choose a captain and deputy. How many choices?
2 marks
$6 \times 5 = 30$ ; roles are different.
30.
How many unordered two-person groups can be chosen from 6 students?
3 marks
Ordered pairs: $6 \times 5 = 30$ .
Divide by 2 because each group appears twice.
15.
List the distinct arrangements of AAB and explain why there are not 6.
3 marks
Swapping identical A letters does not create a different result.
AAB, ABA, BAA:3 arrangements.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Factorial represents decreasing choices

Identical objects and unordered groups

See the method in action.

Arrange different books

Count the first two positions.

Include the remaining positions.

Evaluate the product.

Roles versus a group

Choose the two distinct roles.

Identify the repeated counting for a group.

Divide by the number of orders per pair.

Try it yourself.

Check your understanding

Keep practising