Types of Permutation

In the study of permutation, there are some cases such as:

• Permutation with Repetition
• Permutation without Repetition
• Permutation of Multi-Sets

Let’s explain these cases in detail with solved example as follows:

Permutation With Repetition

This is the simplest of the lot. In such problems, the objects can be repeated. Let’s understand these problems with some examples.

Example: How many 3-digit numbers greater than 500 can be formed using 3, 4, 5, and 7?

Since a three-digit number, greater than 500 will have either 5 or 7 at its hundredth place, we have 2 choices for this place.

There is no restriction on repetition of the digits, hence for the remaining 2 digits we have 4 choices each

So the total permutations are,

2 × 4 × 4 = 32

Permutation Without Repetition

In this class of problems, the repetition of objects is not allowed. Let’s understand these problems with some examples.

Example: How many 3-digit numbers divisible by 3 can be formed using digits 2, 4, 6, and 8 without repetition?

For a number to be divisible by 3, the sum of it digits must be divisible by 3

From the given set, various arrangements like 444 can be formed but since repetition isn’t allowed we won’t be considering them.

We are left with just 2 cases i.e. 2, 4, 6 and 4, 6, 8

Number of arrangements are 3! in each case

Hence the total number of permutations are: 3! + 3! = 12

Permutation of Multi-Sets

Permutation when the objects are not distinct

This can be thought of as the distribution of n objects into r boxes where the repetition of objects is allowed and any box can hold any number of objects.

1^st  box can hold n objects

2^nd box can hold n objects

3^rd box can hold n objects
.                                          . 
.                                          . 
.                                          . 

r^th box can hold n objects

Hence total number of arrangements are,

n × n × n . . . (r times) = n^r

Examples: A police officer visits the crime scene 3 times a week for investigation. Find the number of ways to schedule his visit if there is no restriction on the number of visits per day?

Number of ways to schedule first visit is 7 (any of the 7 days)

Number of ways to schedule second visit is 7 (any of the 7 days)

Number of ways to schedule third visit is 7 (any of the 7 days)

Hence, the number of ways to schedule first and second and third visit is

7 × 7 × 7 = 7³ = 343

Permutation in mathematics is the arrangement of the object in a definite order. Permutation is similar to the combination and the basic difference between permutation and combination is that in permutation the order in which the object is taken is important while the combination is the arrangement of the objects when the order of the objects is not important.

Permutation is represented by the letter, P. For example permuation of set A = {1, 2, 3} when taken two object at a time is, {1, 2}, {1, 3}, {2, 3}, {3, 2}, {3, 1}, {2, 1}. In this article, we will learn about, permutation, its formula, examples, representation, properties, and types of permutation in detail.