Permutation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Permutation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A permutation is an ordered arrangement of objects — the number of ways to choose and order $r$ items from $n$ distinct items is $P(n,r) = \frac{n!}{(n-r)!}$ .

With permutations, order matters — first place and second place are different. Think of ranking students: ABC and BAC are different orderings.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A permutation counts the ways to line up $r$ chosen items from $n$ when the order of the line matters.

Common stuck point: The procedure for permutation is the easy part; the trap is using a permutation when order doesn't matter. Asking "Does swapping two of the chosen items create a different valid outcome?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does swapping two of the chosen items create a different valid outcome?

Worked Examples

Example 1

easy

In how many ways can

3

students be arranged in a line from a group of

7

Answer

P(7,3) = 210

First step

Recall the permutation formula for ordered selections:

P(n, r) = \frac{n!}{(n-r)!}

, where

n = 7

and

r = 3

Full solution

2
Expand the factorial ratio: $P(7, 3) = \frac{7!}{(7-3)!} = \frac{7!}{4!} = 7 \times 6 \times 5$
3
Calculate the product: $7 \times 6 \times 5 = 210$

Permutations count ordered arrangements. Since the order in a line matters (ABC differs from BAC), we use permutations rather than combinations.

Example 2

medium

How many distinct arrangements of the letters in the word MISSISSIPPI are there?

Example 3

medium

Solve for

n

P(n,2) = 72

Example 4

medium

Solve for

n

P(n,3) = 120

Example 5

medium

How many

5

-letter words (real or nonsense) can be formed from

\{A,B,C,D,E,F,G\}

with no repeats?

Example 6

hard

How many ways can

7

people sit in a row if two specific people refuse to sit next to each other?

Example 7

challenge

How many ways can

8

different beads be arranged on a circular bracelet, treating rotations AND reflections as the same?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

How many

4

-digit PIN codes can be formed using the digits

\{1,2,3,4,5,6\}

if no digit may be repeated?

Example 2

medium

A club of

8

students must choose a president, vice president, and secretary. In how many ways can the officers be chosen?

Example 3

easy

Compute

P(5,2)

Example 4

easy

Compute

P(4,4)

Example 5

easy

How many ways can 3 distinct books be arranged on a shelf?

Example 6

easy

Compute

P(6,1)

Example 7

easy

In how many ways can a president and a vice-president be chosen from 4 people?

Example 8

easy

Compute

P(3,3)

Example 9

easy

How many 2-letter ordered codes (no repeats) use letters from

\{A,B,C,D,E\}

Example 10

easy

Compute

P(7,0)

Example 11

medium

How many ways can 5 runners finish 1st, 2nd, and 3rd (no ties)?

Example 12

medium

How many ways can the letters of the word

MATH

be arranged?

Example 13

medium

From 8 people, how many ways to seat 3 in a row of 3 distinct chairs?

Example 14

medium

How many ways can the letters of

LEVEL

be arranged?

Example 15

medium

Solve for

n

P(n,2)=42

Example 16

medium

How many ways can 4 distinct keys be arranged on a straight keyring labeled by position?

Example 17

medium

How many 3-digit numbers (no repeated digits) can be formed from

1,2,3,4,5

Example 18

medium

Compute

\frac{P(6,3)}{P(6,2)}

Example 19

medium

How many 4-letter ordered codes (no repeats) use letters from

\{A,B,C,D,E,F\}

Example 20

challenge

How many ways can 5 people sit around a round table (rotations considered the same)?

Example 21

challenge

How many arrangements of

BANANA

are there?

Example 22

challenge

In how many ways can 4 boys and 3 girls sit in a row so that all 3 girls sit together?

Example 23

easy

Compute

P(8,3)

Example 24

easy

Compute

P(9,2)

Example 25

easy

How many ways can

4

students line up at a counter?

Example 26

easy

Compute

P(10,4)

Example 27

easy

In how many ways can

3

medals (gold, silver, bronze) be awarded to

6

runners?

Example 28

medium

How many distinct arrangements of the letters in the word

\text{ALABAMA}

are there?

Example 29

medium

How many distinct arrangements of the letters in

\text{COMMITTEE}

are there?

Example 30

medium

How many ways can

6

different paintings be hung in a row on a wall?

Example 31

medium

From a class of

10

, how many ways can a president, vice president, secretary, and treasurer be chosen (all distinct)?

Example 32

medium

Compute

\frac{P(7,3)}{P(7,2)}

Example 33

medium

How many distinct arrangements of the letters in

\text{BOOKKEEPER}

are there?

Example 34

hard

In how many ways can

6

people be seated around a round table (rotations equivalent)?

Example 35

hard

In how many ways can

5

different books be arranged on a shelf if

2

specific books must always stand together?

Example 36

hard

How many distinct arrangements of

\text{TENNESSEE}

are there?

Example 37

hard

How many ways can

3

boys and

3

girls stand in a row if they must alternate (boy-girl-boy-girl-boy-girl or the reverse)?

Example 38

hard

A license plate has

3

letters (no repeats, from

26

) followed by

3

digits (no repeats, from

0

–

9

). How many such plates exist?

Example 39

challenge

How many

4

-digit numbers can be formed from the digits

\{0,1,2,3,4,5\}

with no repeated digit?

Example 40

challenge

In how many ways can the letters of

\text{ARRANGE}

be arranged so that the two

R

's are NOT adjacent?

← Back to Permutation Formula Explained Practice Mode

Related Concepts

Factorial Combination

Background Knowledge

These ideas may be useful before you work through the harder examples.

factorial