Permutations vs Combinations | Defintion, Formulas, Difference & Examples

01 May 2025 985 Views Share

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Mathematics is not for the faint of heart. It takes a genius-level brain and a penchant for numbers and formulas to ace your paper. Here is an entertaining attribute about mathematics: If you multiply 111,111,111 by 111,111,111, you will acquire a palindromic numeral of 12,345,678,987,654,321. Interesting, right? Well, this subject is full of surprises, and in this blog, we will uncover one such fun concept: permutation vs. combination.  

You will also discover their formulas and examples that will help you understand how it is calculated. Along with it, you will also peruse the differences and similarities between the two, and you will also get information regarding when to use them.

Permutations vs Combinations | Definitions

The first thing that you will learn in this blog is the definitions of the terms permutation and combination. So, let's begin with the term permutation.

Permutations

Permutation positively represents the structure of items in a clear order. This concept is essential to understand for instances where the sequence and proper order of objects matter.

In terms of maths, permutation is the total ways in which an arrangement of n objects taken r at a time, considering a sequence.

For example, you may arrange your school books in your wardrobe or decide how many orders of letters are possible in a word. For every new arrangement, you have a new permutation.

You use this concept to calculate the chances of different outcomes in scenarios with multiple trials or arrangements.

Combinations

Combination is unlike combinations. You use permutation when arranging some objects in a specific order. Its only interest is to pick objects out of a vast set, and the order in which they are arranged does not matter. However, the combination means all the ways of picking r objects from n objects without regard to the order.

For example, if a person intends to pick three fruits out of five, he is not interested in the order but the combination.

So, you saw the definition of both the terms. If it is still not clear to you, you can seek an assignment writing service. Now, we will see the formulas of permutation vs combination.

Learning the Formulas of Permutations vs Combinations

In the previous section, you saw the meaning of permutations vs combinations. However, to better understand them, we will now see their formulas and see with examples how they help us solve problems and equations in real life. So, without further ado, let us look at the formulas of permutations vs combinations. 

Permutation

For Example:

If you have five structure and you want three on the stand, the permutation would be:

P (5, 3)=      5!    = 5*4*3*-2*1 = 60

                                  (5-3)!        2*1

Therefore, you saw there are 60 ways in which you can choose which three toys you want to keep on the shelf.

Combinations

For example:

If you have 5 flavors of ice cream and you want to pick 3 out of 5, the combinations would be:

C(5,3)=        5!     =   5*-4*3*2*1   = 10

                               3!*(5-3)!     (3*2*1)*(2*1)

Therefore, you saw there are 10 ways in which you can pick 3 flavors of ice cream for yourself.

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Difference Between Permutation vs Combination

Up until now, you saw the definition and formulas of permutation and combination. in this section, you will encounter the differences between the two concepts. You will be able to better understand them with the help of distinction. So, let us see the differences between combinations vs. permutations. 

So, you also saw the differences between permutation and combination and we hope it was clear from the above table. You may be asked to write a research paper on this topic, and if you have any concern, you can seek research paper help.

Similarity Between Permutations vs Combinations

Now you also know the difference between the two but do you know there are similarities between combinations vs. permutations? Yes! you heard it right and it this section we will discover those in detail. So, lets begin with it.

Yes, no doubt there are commonalities between permutations and combinations as they share the ground of counting the number of ways to select and arrange components from a set. The key difference between the two is in permutation, order of selection is important, however, in combinations, the order of selection does not matter.

The similarities between the two are:

• Both combinations vs permutations involve choosing items from a set.
• In a technical sense, combinations are considered as a subset or subsidiary of permutations. You can consider combinations as a permutations where you have discarded the sequence from the arrangement.
• You use both permutations and combinations in statistics, probability and various other fields where counting and arranging possibilities are paramount.

You utilize permutations and combinations in accounting also.If you want to get help choosing a topic under this subject, you can infer accounting research topic for help.

When to Use Permutations vs Combinations

We have come so far and have learned many things. You saw the difference, similarity and other vital things related to combinations vs. permutations. Now, you will see when to use these concepts. So, without any pause, let's start!

Permutations and combinations are more than just maths phrases; they also play vital roles in real-life cases.

Permutations

• They are all about sequence and arrangement.
• Whenever you are required to assemble or rank objects or things where sequence matters, you are dealing with permutations.
• Example: When deciding where to make people stand in a line or organize a book on the help.
• They are helpful in instances like seating arrangements, making schedules or even deciding in what order people will finish a race.

Combinations

• On the other hand, in combinations, your complete focus is on choosing objects without worrying about the order.
• It is most useful when you have to pick groups or subsets without thinking about the sequences.
• They are more helpful in situations like choosing members for a team, picking dishes for a meal plan, lottery tickets, etc.
• These concepts help you to simplify your decisions and solve your issues. Moreover, they turn complex situations into manageable calculations.

So, you saw when to use permutations vs. combinations in real-life situations with the examples given above.

Understanding Permutations vs Combinations Through Examples

Although we have provided you with examples about this concepts in various places, we have dedicated this section to give you examples of both the principles. So, let us see the permutations vs. combinations examples.

Permutation

1. How many possible arrangements are there in which you can make 4 people stand in a row 

Solutions:

Here, n= 4 and r= 4, because we are arranging all four people.

P(4,4)=     4!    =  4*3*2*1  = 24

                                  (4-4)!        1

So, there are 24 ways in which you can make people stand in a row.

2. Pick a prime minister, VP, and secretary from a group of 10 people.

P(10,3)=  10!  =  10*9*8= 720

                                    7!         

Combination

1. Choosing a team of three people from a group of 10

C(10,3)=    10!     = 10*9*8*7*6*5*4*3*2*1  = 120

                          (7!*3!)             3*2*1

2. You have 6 students and you need to choose 2 to represent the entire class.

Solution:

Here, n= 6 and r= 2

C(6,2)=     6!      =  6*5*4*!  =   6*5  = 15

                            2!*(6-2)!      2*1*4        2*1

So, these are the examples of permutations vs combinations. We hope both these examples will be enough to understand these principles properly.

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So, you saw in the entirety what permutations vs combinations are, along with examples and formulas. We can understand that understanding math principles can be tricky and not everyone's cup of coffee. 

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