Permutations | Year 11 | Ascent Library

01 / LEARN

The idea, step by step

An ordered selection uses different roles or positions

A permutation is an ordered selection of distinct objects. Choosing first, second and third place differs from choosing a three-person team: swapping two people changes the result. For r positions from n distinct objects without repetition, multiply n choices, then n−1, until r factors have been used.

n P_{r} = \frac{n!}{(n - r)!}

Build restrictions into the counting model

If two named objects must be adjacent, regard the pair as one block, arrange the blocks, then count the orders inside the pair. If repeated symbols are indistinguishable, ordinary n! overcounts: divide by the factorial of each repeated group. State whether a rotation changes an arrangement; this lesson uses labelled positions or a straight row unless stated otherwise.

A common trap

Dividing by $r!$ even though the positions are different.

Instead: A permutation already counts different orders as different outcomes; divide only when order is irrelevant.

02 / WORKED EXAMPLES

See the thinking

Example 1

Award different medals

Nine finalists receive gold, silver and bronze, with no ties. Count the outcomes.

The medal roles are different.
$9 P_{3} = 9 \times 8 \times 7 = 504$ .

504.

Example 2

Adjacent books

Five different books include two named novels that must stand together. Count the rows.

Treat the two novels as one block: 4 units can be arranged in $4!$ ways.
The novels can switch places in 2 ways.
$4! \times 2 = 48$ .

48.

Check your understanding

Choose a president and secretary from 6 people. How many outcomes?

Show answer

30.

The two roles differ, so $6 \times 5 = 30$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Count ordered selections of 3 different letters from A, B, C, D, E.
2 marks
Seven runners compete for first and second. Count the ordered results.
2 marks
How many rows of 6 distinct students have Ava first?
2 marks
How many rows of 6 distinct students have Ava and Ben adjacent?
3 marks
Count distinct arrangements of the letters in LEVEL.
3 marks
From 8 people choose 3 ordered speakers. One particular person must speak first. Count the schedules.
2 marks

Read the worked solutions

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

Count ordered selections of 3 different letters from A, B, C, D, E.
2 marks
$5 P_{3} = 5 \times 4 \times 3 = 60$ .
60.
Seven runners compete for first and second. Count the ordered results.
2 marks
$7 \times 6 = 42$ .
42.
How many rows of 6 distinct students have Ava first?
2 marks
Fix Ava, then arrange the other 5: $5! = 120$ .
120.
How many rows of 6 distinct students have Ava and Ben adjacent?
3 marks
Arrange the pair block and 4 others in $5!$ ways.
Multiply by 2 orders of the pair: $240$ .
240.
Count distinct arrangements of the letters in LEVEL.
3 marks
There are 5 letters with L repeated twice and E repeated twice.
$5! / (2! 2!) = 30$ .
30.
From 8 people choose 3 ordered speakers. One particular person must speak first. Count the schedules.
2 marks
Fix that person first.
Choose the next two in $7 \times 6 = 42$ ways.
42.

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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