Combinatorics – the field of Mathematics that deals with counting, arrangements, permutations, and combinations – is often one of the most confusing areas. However, it forms the basis of the entire domain of Probability and eventually plays a crucial role in Machine Learning and Artificial Intelligence. Because of these reasons, Permutations and Combinations is a topic that needs to be mastered before proceeding further.
One of the primary confusions that act as a roadblock is the difference between permutations and combinations. For that reason, we’ll take an in-depth look at the key definitions and features of Permutations and Combinations. This will explain how both these terms differ and which one should be applied in which scenario.
Let’s begin!
What are Permutations and Combinations – The Differences Between Them
Let’s try to understand these crucial terms using some examples. Suppose you want to order a salad for lunch. Your preferred salad may be a mixture of tomatoes, carrots, radishes, and beetroot. Now, you don’t care about the order in which these individual veggies are added to your salad as long as all of them are there. All you care about is having all the required vegetables in your salad bowl. The salad could consist of “tomatoes, carrots, radishes, and beetroot” or “tomatoes, carrots, beetroot, and radish”. Both the scenarios will ideally be the same for you – as a salad consumer.
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Starting with Permutation
Now, let’s change the example a bit and think about your Debit Card PIN. If your PIN is 7986, it is a collection of digits 7, 8, 9, and 6. However, in this case, not all arrangements of these digits will end up being your pin. It’s just one specific sequence – 7896 – that is your PIN. In this case, the order is essential.
Permutations are precisely like your PIN details – the order is extremely important. Details are important for permutations. To a permutation, 6/8/9 is entirely different from 9/6/8, which is different from 8/6/9 and so on. For permutations, therefore, the order of entities must be preserved at all costs.
So, to define it in a bit more technical sense – Permutation is a process of selecting different items where the order of selection matters. It can be described as the number of ways of arranging some or all items of a given set.
For example, consider a set – {a, b, c}. In this, all the permutations of the elements are as follows:
- abc
- acb
- bac
- bca
- cab
- cba
Special Cases of Permutations
There are two special cases of Permutations that you should keep in mind:
1. With Repetition
Permutations for ‘k’ of something from total ‘n’ different types can be said to be n*n*n*…k times.
The reason for this is simple – when a thing has n different types … you have ‘n’ number of choices each time.
For example: choosing 3 of those things, the permutations are:
n × n × n
(n multiplied 3 times)
More generally: choosing ‘n’ of something that has ‘k’ different types, the permutations are:
n × n × … (k times)
2. Without Repetition
Without Repetition, the choices will not remain ‘n’ each time. Instead, the values keep decreasing with each choice you make. Here is an example to understand this better:
Try to think of the number of different 4-card hands made from a deck of cards?
Now, for the first card, you have an option of selecting any 1 of 52 cards. So, you have 52 choices. Once you’ve made your first choice, you can’t pick the same card again, so the choices for the next slot become 51. Likewise, every next draw will result in fewer choices from you than earlier. This formula can be generalised as:
To generalise this, the formula for the different permutations of ‘k’ different objects from a group of ‘n’ different objects can be given as:
P(n,k) = nPk = n! / (n−k)!
Where nPk is the number of permutations of ‘k’ different objects from a set of ‘n’ different objects, and n! = n*(n-1)*(n-2)*(n-3)*…. .
Moving on from Permutations – now to Combinations
A Combination can be understood as a technique for determining the number of different possible arrangements in a set of different elements – where the order of selection is not relevant. In combination, you can select the items in any order – remember our earlier example of your salad bowl.
Therefore, the combination is simply the way of selecting different items from a bulk collection so that the order is not important. To understand this better, take the following example:
Suppose we have three digits – 1, 2, 3 – and we want to make a three-digit number. The possible numbers are 123, 213, 132, 231, 312, and 321. Using combinations, we can find the number of ways in which 1, 2, 3 can be placed in a particular order more easily. A combination is the selection of k things from a collection of n things without any replacement and can be written mathematically in the following manner:
C(n,k) = nCk = n! / k! * (n−k)!
Let’s understand this formula better using an example. Try to find out the number of ways in which a coach can choose three swimmers from a group of 6 swimmers.
Using the formula:
nCk = n! / k! * (n−k)!
In our question, the value of n is 6, and the value of k is 3. Keeping that in the formula, we get:
C(6,3) = 6! / 3!*2! = 60 => The coach can choose 3 swimmers from a set of 6 swimmers in 60 different ways.
Some Common Examples of Permutations and Combinations
Let’s look at some day-to-day examples to help you understand the differences between permutations and combinations in a better manner. Through these examples, you’ll be able to spot the differences between these two techniques easily.
1. Permutations
- Arranging different people, numbers, alphabets, digits, vegetables, or colours.
- Selecting a team captain from a team of 11 players.
- Picking three favourite colours from several different colours.
- Selecting first, second and third winners.
2. Combinations
- Selecting the food menu, clothes from a list, subjects for courses, etc.
- Picking different numbers of people from a group of people.
- Picking two colours from a colour book.
- Picking four winners only.
Relation Between Permutation and Combination
Permutations and Combinations essentially refer to the different ways in which objects from a set may be selected – both with or without Repetition – to form new subjects. So, both of these concepts can be understood as counting the number of subsets for a given set. This selection of subsets is called permutation when the order of selection is important and a combination when order is not that important.
In a more mathematical sense, permutation and combination are closely related to one another. Combination is simply the counting of different selections that can be made from n objects. On the other hand, permutation is counting the number of different arrangements from n objects.
If you look closely at the below two formulas of Permutation and Combination, you’ll be able to derive a mathematical relationship between the two on your own. Check it:
- nPr = n!/(n-r)!
- nCr = n!/[r! (n-r)!]
=> nPr = nCr / r!
=> nCr = r! * nPr
The equation mentioned above is the mathematical relationship between permutation and combination.
Difference between Permutation and Combination
Here is a table that will make the basic differences between permutation and combination easier to understand.
| Basic difference between permutation and combination | |
| Permutation | Combination |
| A permutation is used if you want to arrange a set of elements into a sequential order/arrangement. | A combination is used when you want to find out the maximum number of groups that can be formed from a large set of elements where the sequence is not considered. |
| Sequence order is an important consideration. | Sequential order is not important, while choice is. |
| Permutation refers to the arrangement of elements. | The combination does not consider any particular arrangement of elements. |
| You can find out several permutations from a single combination. | You can only find out a single combination from a single permutation. |
| Ordered sets are called permutations. | Unordered sets can also be called combinations. |
| Formula for permutation:
P(n,k) = nPk = n! / (n−k)! Where nPk refers to the number of permutations of k distinct elements chosen from a set of ‘n’ different objects. | Formula for combination:
nCk = n! / k! * (n−k)! Where nCk refers to the number of possible combinations of ‘k’ different objects chosen from a set of ‘n’ different objects. |
Let’s understand the difference between permutation and combination with example to give you an idea of how we use them in real life.
- Forming a team for a game: We often use combinations to determine how many possible teams could be formed from a large group of players to ensure a fair distribution.
- Seating arrangements for an event: You can use the permutation formula to determine the number of possible seating arrangements for formal events or official seating plans.
- Combination while forming committees: You can apply combinations to find out the possibility of forming a committee by selecting a few individuals from a larger group.
- Creating a password: We can also use permutations to calculate the number of possible passwords that can be formed using a given set of numbers, symbols, and alphabets.
Points to Remember
- The combination is the number of ways you can select a subset of objects from a larger set without taking the order into consideration. While permutation is the different number of ways you can arrange a set of objects in a specific order.
- If n and k values are the same, then the number of permutations will always exceed the number of combinations.
- Since the order does not matter while calculating combinations, the result of choosing the same k objects from a set of n elements will always be the same.
- Since the order is essential in permutation, even if you choose the same k objects from a set of n objects, the result will differ depending on the selection order.
In Conclusion
With that, we come to the end of this blog post on the differences between Permutation and Combination. Keep in mind that the field of Combinatorics is exceptionally vast and sets the base of many other vital fields of maths – especially when it comes to applicative fields like Probability or Machine Learning. What we have discussed in the article is just the fundamental differences between Permutation and Combination. However, with this knowledge by your side, you can easily tackle all the confusions that generally students face while solving problems around PnC.
If you understood everything in this article, we recommend you to dive deeper and acquaint yourself with other nuances of combinatorics. If you didn’t understand the article too well – please ask your doubts in the comments below.
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