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


On this page...

Arranging n objects
Number of Permutations
Permutations of different kinds
Arranging in a circle
Exercises

Example

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


Answer


Reminder - Factorial Notation

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

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

Each of the theorems in this section use factorial notation.


Theorem 1 - Arranging n Objects

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


Example

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


Answer


Theorem 2 - Number of Permutations

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

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Notes:

(1) Pnn = n! (since 0! = 1)

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

nPr

and others have:

nPr


Example

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


Answer


Example

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?


Answer


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

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Example

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


Answer


Theorem 4 - Arranging Objects in a Circle

There are (n - 1)! ways to arrange n distinct objects in a circle.


Example

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


Answer


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?


Answer


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?


Answer


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?


Answer


Exercise 4

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


Answer



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