What are Permutation and Combination? – Formulae, Example, Derivation

Permutation and combination are methods for representing a collection of things by picking them from a set and constructing subsets. It specifies numerous methods to arrange a specific set of data. Permutations are when we choose data or objects from a certain group, whereas combinations are the order in which they are represented. Both notions are crucial in mathematics. 

What is Permutation?

A permutation is a specific arrangement of objects. The numbers or elements are arranged in a sequence or linear order here. For example, the permutation of set A= {1,4} is 2, as in {1,4} {4,1}. There are no other ways to arrange the items in the set. 

What is Combination?

A combination is a method of choosing things from a collection in which the order of selection is irrelevant. Assume we have three suits. The number of ways we may choose two numbers from each suit is thus specified by combination. 

It is feasible to count the number of combinations in smaller circumstances, but the set of combinations increases with the number of groups of components or sets. 

Permutation and Combination Formula

Mathematics contains several permutation and combination formula aptitudes. Yet, the majority of these permutation combination formulae are founded on two fundamental concepts. Here they are: 

Permutation Formula

If the total quantity of data is “n” and the option is “r,” then the permutation will be (without replacement and concern for order)- 

permutation formula

Combination Formula

The selection of “r” items from a batch of “n” data without consideration for order or replacement- 

Combination Formula

These are the essential formulae for calculating probability permutations and combinations. 

Moreover, the relationship between the two is

permutation and Combination Formula

Difference Between Permutation and Combination



Permutation refers to the choosing of things in which the sequence of selection is important.  The term “combination” refers to the selection of things in which the sequence of selection does not matter. 
Digits, letters, numbers, people, alphabets, and colors are arranged  Menu, cuisine, clothing, subjects, and team selection 
Choose a team captain, pitcher, and shortstop from a pool of candidates  Choose three team members from a large group 
Choose two favorite colors from a color brochure in sequence.  Choose two colors from a color catalogue 
Choosing the first, second, and third finishers  Choose three finishers 

Advantages of Permutation and Combination

Permutation – a list of data (where the order of the data is important). 

Combination – a group of data (where the order of the data is irrelevant). 

Finding permutation and combination eases your real life with its applications in many various industries. Its applications are vast, a few are mentioned below. 

Applications of Permutations and combinations in real-life

  • Combination lock  
  • Passwords 
  • Phone numbers  
  • Car plate numbers  
  • Playing the piano  
  • Making a cup of coffee 
  • Picking three finalists  
  • Selecting 2 out of 5 questions 
  • Selecting teams  
  • Clothes combination 

Final Notes

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What are the 4 types of permutation? 

  • Permutations with repetition 
  • Permutations without repetition 
  • Permutations with multi-sets 
  • Circular permutations 

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