3. Permutations (Ordered Arrangements)

An arrangement (or ordering) of a set of objects is called a permutation. (We can also arrange just part of the set of objects.)

In a permutation, the order that we arrange the objects in is important

Example 1

Consider arranging 3 letters: A, B, C. How many ways can this be done?

Reminder - Factorial Notation

Recall from the Factorial section that n factorial (written `n!`) is defined as:

n! = n × (n − 1) × (n − 2) ... 3 × 2 × 1

Each of the theorems in this section use factorial notation.

Continues below

Theorem 1 - Arranging n Objects

In general, n distinct objects can be arranged in `n!` ways.

Example 2

In how many ways can `4` different resistors be arranged in series?

Theorem 2 - Number of Permutations

The number of permutations of n distinct objects taken r at a time, denoted by `P_r^n`, where repetitions are not allowed, is given by



(1) `P_n^n=n!` (since `0! = 1`)

(2) Some books use the following notation for the number of permutations:


and others have:

`{::}^n P_r`

Example 3

In how many ways can a supermarket manager display `5` brands of cereals in `3` spaces on a shelf?

Example 4

How many different number-plates for cars can be made if each number-plate contains four of the digits `0` to `9` followed by a letter A to Z, assuming that

(a) no repetition of digits is allowed?

(b) repetition of digits is allowed?

Theorem 3 - Permutations of Different Kinds of Objects

The number of different permutations of n objects of which n1 are of one kind, n2 are of a second kind, ... nk are of a k-th kind is

`(n!)/(n_1!xxn_2!xxn_3xx...xx n_k!`

Example 5

In how many ways can the six letters of the word "mammal" be arranged in a row?

Theorem 4 - Arranging Objects in a Circle

There are `(n - 1)!` ways to arrange n distinct objects in a circle (where the clockwise and anti-clockwise arrangements are regarded as distinct.)

Example 6

In how many ways can `5` people be arranged in a circle?


Exercise 1

In how many ways can `6` girls and `2` boys be arranged in a row

(a) without restriction?

(b) such that the `2` boys are together?

(c) such that the `2` boys are not together?

Exercise 2

How many numbers greater than `1000` can be formed with the digits `3, 4, 6, 8, 9` if a digit cannot occur more than once in a number?

Exercise 3

How many different ways can `3` red, `4` yellow and `2` blue bulbs be arranged in a string of Christmas tree lights with `9` sockets?

Exercise 4

In how many ways can `5` people be arranged in a circle such that two people must sit together?

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Math Lessons on DVD


Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.