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

`P_r^n` `=n(n-1)(n-2)...(n-r+1)` `=(n!)/((n-r)!`

Notes

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

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

`nPr`

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?

Exercises

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?