Permutations and Combinations

Calculation of 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 word, 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 permutation formulae. In most mathematics field, 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 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 diff between permutation and combination is that combinations are different ways of selection without regarding the sequence. And permutations are various ways of arrangement with 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 clear 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

  1. 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

  1. 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.

FAQ (Frequently Asked Questions)

1. How Do You Solve Permutation And Combination Card Problems?

Let us discuss it with an example. What are the ways to draw 2 cards in a way that one is a red face card and another is a black card? Now, you know that a pack has 52 cards, in which there are 4 suits, and each suit has 13 cards. Among these suits, there are 2 red and 2 black suits. Thus, there are 6 red face cards and 26 black cards. So, the solution will be – drawing one red face card = 6C1 = 6! / {1! (6-1)!} = 6 and drawing one black card – 26C1 = 26! / {1! (26-1)!} = 26. Hence, the total way of drawing = 6 X 26 = 156.

2. How Do You Solve For Permutations And Combinations Cat Questions?

Initially, this chapter may seem a little harder to understand. However, once you learn and understand the formulas, you can solve all types of questions. For best result, you can solve the questions from previous years. Moreover, you also opt for online permutation and combination chapter-wise mock test. This way, you can practice problems and increase your accuracy and confidence.

3. What Is The Way Of Calculating Permutation?

While expressing permutation, you write nPr. In this expression, n denotes the number of data or thing to select, P signifies permutation, and r refers to the number of items to select. To find out permutation, you use this formula of (n!) / (n-r)!. From the given data, once you out the digits in the place of n and r, you can find out the permutation number easily.

4. What Is The Method For Calculating Combination?

By calculating combination, you can find out the entire outcome of an arrangement without acknowledging the order of possible outcomes. You use the formula of nCr = (nr) = nPr / r! = n! / {r! (n-r)!}. In this formula, n is the total item number and r is the item number you are going to choose at one time. Now, as per the question, you can put the data and calculate the combination number quickly.