Permutations and Combinations

Permutation and Combination are the most basic concepts in mathematics, and with these concepts, a new branch of mathematics is introduced to students, i.e., combinatorics. Permutation and Combination are the methods of arranging a group of objects by choosing the objects in a particular order and creating their subsets. Choosing the data or objects from a particular group is known as permutation, while the order in which they are arranged is called a combination.

Permutation

Permutation is the different ways of looking at the components provided and carrying them one by one, or some, or all at a time. To clarify, if we have two components, A and B, then there are two possible performances, AB and BA.

A numeral of permutations when ‘r’ components are positioned out of a total of ‘n’ components is nPr. For instance, let n = 3 (A, B, and C) and r = 2 (All the permutations of size 2 are possible). Besides, there are 3P2 such permutations, which is equal to 6. These six permutations are AB, AC, BA, BC, CA, and CB. The six permutations of A, B, and C, taken three at a time, are shown in the image added below: 

Permutation Formula

Permutation formula is used to find the number of ways to pick r things out of n different things in a specific order, and replacement is not allowed and is given as follows:

nPr=(n−r)!/n!

Explanation of Permutation Formula

We know that permutation is the arrangement of r elements out of n where the order of the arrangement is important (AB and BA are two different permutations). When there are three different numerals, 1, 2 and 3, and you want to permute the numerals taking two at a moment, it shows (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), and (3, 2). Thus, it can be done in 6 ways.

Here, (1, 2) and (2, 1) are different. Moreover, if these three numbers are divided into the handling of all at one time, then the interpretation will be (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2) and (3, 2, 1) i. e. in 6 ways.

Usually, the number of ways to choose n distinct things and take r (r < n) at a time is n(n – 1)(n – 2)…(n – r + 1). To begin with, the first thing can be any of the n things. Hence, the second thing will be one of the remaining n – 1 things after picking the first thing. The third thing can be any of the rest of the n – 2 things. Similarly, the rth thing can be any of the remaining n – (r – 1) things.

Thus, the total number of permutations of n distinct things carrying r at a time is n(n – 1)(n – 2)…[n – (r – 1)], which is written as n Pr. Or, in other words,

nPr=(n−r)!n!​

Combination

It is the different parts of a common number of components taken one at a time, or some, or all of them at once. For instance, if there are two components, A and B, then there is only one way to choose two things: choice of both of them.

For instance, let n = 3 (A, B, and C) and r = 2 (All combinations of size 2). After that, we have 3C2 such combinations, which is equal to 3. These are the three possible combinations: AB, AC, and BC.

In this case, the Combination of any two letters out of the three letters A, B, and C is given below; we observe that in the Combination, the order in which A and B are taken is not important because AB and BA are the same Combinations.

Note: In the same example, we have different points for permutation and Combination. To put it simply, AB and BA are two different items, i.e., two different permutations, but for selection, AB and BA are the same, i.e., the same Combination.

Combination Formula

Combination Formula is used to choose ‘r’ components out of a total number of ‘n’ components, and is given by:

nCr​=n!/r!(n−r)!​

Through the above formula for r and (n-r), we obtain the same result. Thus,

nCr=C(n-r)

Explanation of Combination Formula

Combination, in contrast, is the type that comprises a bunch. Moreover, if the sets are to be created with the usage of two numbers only, then the possible combinations are (1,2), (1,3) and (2,3).

This is the case of (1, 2) and (2, 1), which are equally distinct, unlike the permutations in which they are different. This can be expressed as 3C2. Generally, the number of combinations of r items of n indistinguishable things taken n at a time is.

nCr= n/!r! * (n-r)!=nPr/r!

Derivation of Permutation and Combination Formulas

These formulae on Permutation and Combination can also be deduced through the basic counting methods as they represent the same thing. Derivation of these formulas is as follows:

Derivation of Permutations Formula

Permutation means selecting r distinct objects from n objects without repetition, and the order of selection is essential. Using the fundamental theorem of counting and the definition of permutation, we get

P (n, r) = n . (n-1) . (n-2) . (n-3).  . . .  .(n-(r+1))

By Multiplying and Dividing the above with 

 (n-r)! = (n-r).(n-r-1).(n-r-2).  . . .  .3. 2. 1

we get,

P (n, r) = [n.(n−1).(n−2)….(nr+1)[(n−r)(n−r−1)(n-r)!] / (n-r)!

⇒ P (n, r) = n!/(n−r)!

Thus, the formula for P (n, r) is derived.

Derivation of Combinations Formula

The Combination is a selection of r items out of n items when the order in which the selection has been made is of no importance. This formula is determined as:

C(n, r) = Number of Permutations / Number of ways to arrange r of different items.

[Recall that by the fundamental theorem of counting, we know that the number of ways to arrange r different objects in r ways is r!]

C(n,r) = P (n, r)/ r!

⇒ C(n,r) = n!/(n−r)!r!

Thus, the formula for Combination, i.e., C(n, r), is derived.

Conclusion

Permutations and combinations have been proven to be essential instruments across various fields, and they provide a clear line between complicated problems and their solution. In order to design a comprehensive approach that will help you understand and master mathematics concepts, switch to Turito’s online classes. Turito has unique options that include a two-teacher model for instant doubt clarification, gamified educational quizzes, on-demand videos, and AI-driven assessments, making the learning experience complete and fun. You do not have to wait any longer to experience the special kind of learning designed just for you. Visit to Turito right now!

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