Permutations and Combinations

Have you ever noticed that the mobile PIN you use can be drawn in several variations? 

Well, this is one of the examples of permutations and combinations. In layman’s words, a combination is when the order is not important, and permutation is when the order is important. With the help of permutations combinations, you can express a group of data in the form of sets and subsets. 

It refers to the different ways of arranging a specific group of data. Both these concepts are vital not only in your board exams but also in all competitive examinations like CAT, JEE, etc. Thus, you need to understand both concepts and the difference between permutation and combination as well. 

Definition of Permutation and Combination

To start learning about this chapter, you first need to understand permutation and combination definition and relation between permutation and combination.

Permutation

A permutation is when you arrange a set of data in some specific order or sequence. Moreover, if the data is already arranged in order, you can rearrange them by using the permutation formula. In most mathematics fields, permutation occurs. 

Combination

Contrary to permutation, a combination is when you choose data from a group without any order or sequence. If the group of data is relatively lesser, you can calculate the number of possible combinations. 

Let us elaborate these definitions with permutations and combinations examples. For example, you have a group of four letters P, Q, R, and S. Now, in how many ways can you choose three letters from this group. Every probable arrangement can be a combination.

However, the ways you can group P, Q, R, and S together, are permutations. So, PQRS, PRSQ, PSQR, PRQS, etc. are permutations. If permutation and combination meaning are not clear, then try a real-life example.

As per permutation combination examples from real life, you can say that selecting winners like 1st, 2nd, and 3rd is a permutation. And, selecting three winners is a combination.

The Difference in Permutation and Combination

Till now, you have learnt the answer to “define combination and permutation”, and that can help you differentiate permutation and combination. As per their definitions and examples, the major difference between permutation and combination is that combinations are different ways of selection without regarding the sequence. And permutations are various ways of arrangement regarding the order.

This is the key permutation combination difference that you should understand to consolidate the concept.

Basic Formula of Permutation and Combination

Many permutation and combination formula aptitudes are there in Mathematics. However, most of these permutation combination formulas are based on two essential formulas. Here are these permutation and combination basic formulas – 

Permutation Formula

If the total number of data is “n” and the choice is of “r” things, then permutation will be (without replacement and regarding an order)-

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

Combination Formula

From a group of “n” data, the selection of “r” things without regarding order and replacement-

nCr = (nr) = nPr / r! = n! / {r! (n-r)!} 

These are the key formulas to find out probability permutations and combinations. Moreover, the relation between these two is nCr = nPr / r!.

Now, let us solve some permutation and combination questions to clean out your doubts.

Permutation and Combination Word Problems

By solving the following permutation and combination problems, you can understand how to derive these formulas for permutation and combination NCERT solutions.

1. How to calculate the number of combination and permutation if n = 14 and r = 3

Class 11 permutation and combination solutions:

As per the question, n = 14

r = 3

By deriving the permutation formula-

nPr = (n!) / (n-r)! = 14! / (14 - 3)! = 14! / 11! = (14 X 13 X 12 X 11!) / 11! = 2184

Now, from the combination formula-

nCr = (nr) = nPr / r! = n! / {r! (n-r)!} = 14! / 3! (14 - 3)! = 14! / 3! (11!) = 14 X 13 X 12 X 11! / 2! X 11! 

• 1092

2. How many 4- digit numbers can you form from 1, 2, 3, and 4 –

1. With repetition?

2. Without repetition?

NCERT Class 11 Permutation and Combination Solution:

As there will be a 4-digit number, then let the digit be ABCD. Here, D is the unit place, C is the 10th place, B is 100th place, and A is at thousand place. 

1. Now, with repetition, at the place of D, the possible numbers of the digit are 4. Also, at the 

place of A, B, and C, the probable number of digits are 5.

So, the total possible 4-digit numbers are – 4 X 4 X 4 X 4 = 256

2. The possible number of digits at the place of D is 4; hence it is the unit place. Now, 

Without repetition, one digit is occupied at D. So, for place C the possible digit will be 3 and there will be 2 possible digits for B and 1 for A.

Hence, the total possible 4-digit numbers without repetition are – 4 X 3 X 2 X 1 = 24.

From the above permutation and combination questions with solution, you must have understood the pattern of questions which can come in your examinations.

Nonetheless, if you need some more NCERT solutions permutation and combination, you can go to our Vedantu website and check all study materials on permutation and combination answers. These are also accompanied with questionnaires and exercises. Furthermore, you can also learn permutation and combination online from our online sessions. 

Get our Vedantu app now for updated NCERT solutions class 11 permutations and combinations.

The different concepts where permutations and combinations are used will help us differentiate between the two. Permutations and Combinations help us to get a group of data in the form of sets and subsets.and can also be defined as Different ways of arranging specific groups of data.

Permutation is used when the objects and things are of different kinds. The smaller groups that can be formed from the elements of a larger group is Combination.

When we need to arrange a sequence of things, we need Permutations whereas in order to find  how many possible groups can be formed , we need Combinations.

For a list of data, where the order of data is important, we have Permutation and A group of data where we do not need any order, is Combination.

 Permutation and Combination formulas help us in understanding the calculations  of the difference between these two. The two important formulas are

Permutation is a choice of things ‘r’, from a set of things ‘n’ without any replacement and also where an order matters :

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

Combination is a choice of things ‘r’, from a set of things ‘n’ without any replacement but where order is not required and does not matter :

nCr = (nr) = nPr / r! = n! / {r! (n-r)!} 

Let us take the help of an example from the NCERT Solution:

Q. How many 4- digit numbers can you form from 1, 2, 3, and 4 –

1. With repetition?

2. Without repetition?

NCERT Class 11 Permutation and Combination Solution:

As there will be a 4-digit number, then let the digit be ABCD. Here, D is the unit place, C is the 10th place, B is 100th place, and A is at thousand place.

1. Now, with repetition, at the place of D, the possible numbers of the digit are 4. Also, at the place of A, B, and C, the probable number of digits are 5.So, the total possible 4-digit numbers are – 4 X 4 X 4 X 4 = 256.

2. The possible number of digits at the place of D is 4; hence it is the unit place. Now, without repetition, one digit is occupied at D. So, for place C the possible digit will be 3 and there will be 2 possible digits for B and 1 for A.Hence, the total possible 4-digit numbers without repetition are – 4 X 3 X 2 X 1 = 24.

From the above two examples from the NCERT  Question and Solutions, we can get a clearer picture regarding Permutation and Combination,and also get a little idea of the pattern of questions we can expect  during the examinations.