Differentiation Between Permutation and Combination

Permutation and combination are often confused with each other due to the minute differences between them. Let us try to understand them both with examples before understanding the permutation and combination differences.

Definition of Permutation:

Permutation, in simpler terms, can be defined as a mathematical calculation used to find several ways a set can be arranged. There is one more distinct property to permutation, i.e., the order of the arrangement matters in permutation. 

For example, if we want to arrange two objects, A and B, present in a set C, then as per permutation, it can be done in only two ways, i.e. AB and BA. Permutation can be denoted mathematically by the below-mentioned formulae for n objects in which k selections need to be made:

P (n,k) = n!/(n-k)!

where n! represents “n factorial”, which is the multiplication of all the positive integers below it till 1. For example, the factorial of 4 is 4*3*2*1, equal to 24. Factorial of 0 is 1.

Based on the above definition of Permutation, lets us take a few more examples to understand it in more detail:

Example 1: How many outcomes are possible if pair of objects need to be selected from five objects such as P, Q, R, S and T.

Solution: We know k selections in n objects can be given by the below-mentioned formulae i.e.

P (n,k) = n!/(n-k)!

where n = 5 and k = 2.

P(5,2)   = 5!/(5-2)!

            = 5*4*3*2*1/3*2*1

            = 20

Twenty outcomes are possible if pair of objects need to be selected from five objects.

Example 2: Consider a meeting room having 4 chairs and 3 people. How many ways, these 3 people can sit on those 4 chairs?

Solution: In this case; n=4 and k=3.

Applying the permutation formula, P (n,k) = n!/(n-k)!

P(4,3)  = 4!/(4-3)!

            = 4*3*2*1/1

            =24

There are 24 ways. These 3 people can sit on those 4 chairs in the meeting room.

Types of permutation

Permutation can be broadly classified into three different categories:

1. Permutation where repetition is not allowed: This can be represented by the formulae P (n,k) = n!/(n-k)!. Example 1 and 2 mentioned above represents this particular type.
2. Permutation where repetition is allowed: This can be represented by the formulae P (n.k) = nk where “n” is the number of objects, and “k” is the selection of the object.

Example 3: How many outcomes are possible if pair of objects need to be selected from five objects such as P, Q, R, S and T. with repetition allowed.

Solution: Permutation with repetition can be represented as nk where “n” is 5 and “k” is 2.

Therefore, P(5.2)        = 52 

                                    = 25.

Twenty-five outcomes are possible if pair of objects need to be selected from five objects with repetition allowed.

Example 4: How many 3 letter words can be made out of the word “Apple” when repetition is allowed.

Solution: Permutation with repetition can be represented as nk where “n” is 5 and “k” is 3.

Therefore, P(5.3)        = 53 

                                    = 125.

One hundred twenty-five words are possible if 3 letter words need to be selected from “Apple”, with repetition allowed.

                                    

1. Permutation of objects that are non-distinct: Above two types represent when “n” objects are distinct. If objects are not distinct; then permutation can be represented by

P = n!/n1!n2!n3!….nn!

Where n represents a total number of objects and n1, n2,nn represents similar objects among n objects.

Example 5: How many words can be formed by using all the letters of different words given below?

1. PERMUTATION
2. KARNATAKA

Solution:

1) The word PERMUTATION consists of  11 alphabets; hence n=11. Also, in the word PERMUTATION, the alphabet “T” is repeated.

From the above definition, T represents “n1”. Also, the number of times T is repeating is two. Therefore n1 =2.

Now, the permutation of non-distinct objects is given by P =n!/n1!

Hence the required number of permutations is

=11!/2!

=39916800/2

=19958400

2) The word KARNATAKA consist of 9 alphabets; hence n=9. Also, in the word KARNATAKA two alphabets are repeated, i.e. “K” and “A”.

From the above definition, K represents “n1”. Also, the number of times K is repeating is two. Therefore n1 =2.

Also, from the above definition, A represents “n2”, and a number of times A is repeating is four. Therefore n2 =4.

Now, permutation of non-distinct objects is given by P =n!/n1!n2!

Hence the required number of permutations is

= 9!/(2!)(4!)

= 9*8*7*6*5*4!/2*(4!)

= 15120/2

=7560

Definition of Combination

In simple words, the combination can be defined as permutation wherein orders of arrangement does not matter. It means the combination is a mathematical way to calculate the number of ways in which objects can be arranged without considering the order of arrangement. 

The combination mentioned formulae represent combination:

C (n,k) = n!/k!(n-k)!

Example 6: How many outcomes are possible if pair of objects need to be selected from five objects such as P, Q, R, S and T, wherein the order of arrangement does not matter.

Solution: We know k selections in n objects can be given by the below-mentioned formulae i.e.

C (n,k) = n!/k!(n-k)!

where n = 5 and k = 2.

C(5,2)   = 5!/2!(5-2)!

            = 5*4*3*2*1/2*1*3*2*1

            = 10

10 outcomes are possible if pair of objects need to be selected from five objects. If we recall example 1, where 20 outcomes were possible as per permutation formulae. Hence, if the order of arrangement is not important, then there is a change in the outcome represented by combination.

Also, it can be concluded that the value of combination will be always smaller than the value of permutation for any given “n” and “k”.

Example 7: For n=10 and k=5, find respective values for permutation and combination?

Solution: For permutation we know that P (n,k) = n!/(n-k)!

Therefore, P(10,5) = 10!/(10-5)!

= 10*9*8*7*6*5!/5!

= 30240

For combination we know that C (n,k) = n!/k!(n-k)!

Therefore, C(10,5) = 10!/5!(10-5)!

= 10*9*8*7*6*5!/5! (5*4*3*2*1)

= 252

This example also proves the value of permutation is always greater than the value of the combination.

 Permutation and Combination Differences 

From the above discussion and examples, now we can differentiate between permutation and combination as mentioned in below table:

Permutation	Combination
Permutation represents all possible ways in which an object can be arranged wherein the order of arrangement matters.	The combination represents all possible ways an object can be arranged wherein the arrangement does not matter.
For given n and k, the value of permutation is always greater than the combination value.	For given n and k, the value of the combination is always smaller than the permutation value.
Permutation is given by formulae P (n,k) = n!/(n-k)!	Combination is given by formulae C (n,k) = n!/k!(n-k)!
Permutation represents arrangement	The combination represents grouping/selection.
The permutation is denoted by nPk or P(n.k)	The combination is denoted by nCk or C(n.k)

Conclusion

Hence, we hope you have gotten a clear idea regarding the concepts of permutation and combination along with its differences from this article.