Both Permutation and Combination are integral parts of counting numbers with logic. Counting solves probability problems; therefore, learning about Permutations and Combinations before learning probability is greatly important. More importantly, you need to know the key differences between these two. Permutation considers the order of members. On the other hand, the order does not matter in Combination. For instance, the orderly arrangement of numbers, objects, or alphabets is known as Permutation, whereas selecting a cluster of the said objects, numbers, or alphabets can be considered a Combination. 

In this article, we will focus on the key difference between Permutation and Combination by defining them and illustrating various examples that will aid in a better understanding of the two separate concepts.

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What is Permutation?

A permutation is the process of selection, keeping order in mind. It is defined as the number of ways a  few or every member in an order can be arranged. Therefore, the term ‘Permutation’ is all about the order of the members in a set.

For example: 

The Permutations of a small set of letters {a, b, c} are as follows:-

abc acb

bac bca

cab cba

The formula for the total of Permutations of k objects taken from a group or a set of n is normally written as nPk.

Formula:

nPk=n!(n−k)!=n(n−1)(n−2)…(n−n+1)(n−k)(n−k−1)(n−k−2)…(n−k−n−k+1)

The two kinds of Permutation are as follows:-

• Permutations with Repetition

Selecting r from a number of an element consisting of n different types, then the Permutations will be:

n×n×…

 (r times)

Similarly, there aren’t any possibilities for the first selection process. Hence, there aren’t any possibilities for the next selection process, which continues multiplying every time.

It is easier to write down by using the exponent of r:

Therefore, nr=n×n×…

 (up to r times)

Thus, the formula is: nr,

Here, n is the total number of elements you need to choose from a set or cluster of elements. We need to choose r from them. It is also important to note that the order is important and that repetition is allowed.

• Permutations Without Repetition

Lack of repetition, the choices will get reduced every time. Let’s look into the easiest and most commonly used example:                

The total number of different hands of the 4-card made from a card deck:-

In this particular problem, the order is irrelevant because it does not matter what order is followed in the selection of the cards. We will start with four lines to represent the 4-card hand. Let us assume that ‘52’ is placed in the first blank out of all 52 cards in the first draw. Once a card is chosen, one card is selected already. Hence one card less will be available for the next draw. Hence, the second blank will give you 51 available options. Also, you will get two fewer cards in the next draw in the deck, leaving you with 50 options. The formula is as follows – 

P(nr)=nPr=n!(n−k)!

The result of using the above formula is given below:-

P(524)=52P4=52!48!

Here, n is the number of objects you must choose amongst a set of elements, and we select r of them. There are no repetitions, and order does not matter here.

Permutation Examples

• Arrangement of digits, alphabets, numbers, letters, people, colors, and the like.
• Selecting a team keeper or captain and a specific one from one group.
• Selecting two best-loved colors from a book of colors in order.
• Picking the winners of the first, second and third positions.

What is Combination?

Combination is the method of selecting items out of a large collection where the selection order is not important. We can simply say that combination is the way of selecting one group by selecting all or some members in the set. It has no specific order that must be followed when combining the elements in a set.

 In relatively smaller cases, it is easier to count the actual total of Combinations. Combination refers to the combination of n number of things that are taken k at one time without repetitions. It is choosing r objects from a particular set of n objects without replacing and not considering an order. There are numerous ways of creating a combination and all of them are correct in their own right. No particular or ‘right’ method has been set to figure out one combination and therefore has been termed as a combination. 

Using the following combination formula, you can easily acquire the combination in any given set.

C(nr)=nCr=nPrr!=n!r!(n−k)!

Down below, we have illustrated an example to elucidate this:-

Let us take three digits (1,2,3) with which we are required to create a three-digit number, Therefore, we can deduce that only the numbers below are possible:-

123, 132, 213, 231, 312, 321..

Combinations provide an easier way to figure out the number of ways “1 2 3” could be put in a specific order, as we’ve seen previously. The answer is:

3! = 3 ×

 2 ×

 1 = 6

The Permutation’s formula has therefore been reprinted to reduce it by the number of ways the objects can be in order.

Combination Examples

• Selecting food, menus, subjects, clothes, teams, etc.
• Selecting three members from a team or a group.
• Selecting two colors from a book of colors.
• Selecting only three winners.

The Key Points of Distinction Between Permutation and Combination

While calculating probability, learning the differences between Permutation and Combination is key to mastering it. The key points of difference have been illustrated in the table below:-

Permutation	Combination
The various methods to arrange a particular object set sequentially are called Permutation.	The various methods of selecting objects from a huge object set that doesn’t consider the order is known as Combination.
The order is important.	The order is not important.
It will denote the object arrangement.	It will not denote the object arrangement.
Various permutations are acquired from one combination.	Only one combination is acquired from one permutation,
They are defined to be ordered elements.	They are defined to be unordered sets.

Examples of When To Use Permutation and Combination

For instance, if we are required to locate a total of samples that are probable of two from the three objects X, Y, and Z, we must understand which method is relevant to this particular problem. Hence, we will need to check if it is necessary to consider the order or not.

If the object order is integral to this problem, it is relevant to permutation. The possible samples will be as follows:

 XY, YX, YZ, ZY, XZ, and ZX. 

In this instance, XY is different from sample YX. YZ is different from sample ZY. XZ is different from sample ZX.

However, if the object order is a mandate, then the problem can be solved via the  combination method where the possible samples will be as follows:

XY, YZ, and ZX.

Similarities Between Permutation and Combination

If we consider mathematical concepts, “Permutation” and “Combination” are related to one another. Counting selections made from n objects is called Combination, whereas counting the total arrangements from n objects is Permutation. We need to remember that combinations emphasize order, arrangement, or placement but mainly on choice.

Conclusion 

It can be easily deduced that Permutation and Combination are integral in the field of statistics, mathematics, research, and our daily life. It is important to note that permutation is always supposed to be higher than combination. If you want to know more about Permutation and Combination, you can learn more about these concepts from upGrad’s top-tier courses. One great course is a Master of Science in Machine Learning and Artificial Intelligence