Get interactive courses taught by top teachers

Permutation and combination are methods for representing a collection of things by picking them from a set and forming them into subsets. It specifies the numerous methods for organising a set of data. Permutations are used for selecting data or objects from a group, whereas combinations are used to represent the order in which they are represented. In mathematics, both concepts are crucial.

Here we will go through the definition, explanation and formulas of the permutation and combination. Then we will solve some of the examples and previous year's permutations and combinations questions for better understanding.

Fundamental principle of counting

Permutations as an Arrangement

Combinations as Selections

Permutation and Combination of n items taken r at a time

Application of Permutation and Combination

Many important concepts are discussed in the chapter Permutation and Combination. These concepts are discussed in detail below.

Permutation

A permutation is an act of placing all the members of a set into a sequence or order in mathematics. To put it another way, permuting is the process of rearranging the components of a previously sorted set. Permutations can be found in practically every branch of mathematics, with different degrees of importance. When different orderings on certain finite sets are explored, permutations commonly occur.

Combination

The combination is a method of selecting elements from a collection in which the order of selection is not important (unlike permutations). In smaller instances, the number of possible combinations can be counted. Combination refers to the combination of n number of things taken k at a time without repetition. Also, there are some combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used.

Permutation and combination concepts involve a lot of formulas. The two key formulas are as follows:

Permutation Formula

A permutation is the selection of r items from a collection of n items without replacement, with the order of the items being important.

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

Combination Formula

A combination is a selection of r items from a set of n items with no replacements and where order doesn't matter.

\[^{n}C_{r} = \frac {n!}{r!(n-r)!} = \frac {^{n}P_r}{r!}\]

Since a permutation involves selecting r distinct items without replacement from n items and order is important, by the fundamental counting principle, we have

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

This can be written as:

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

Multiplying and Dividing (1) by (n-r) (n-r-1) (n-r-2)........... 3. 2. 1, we get

P (n, r) = $\dfrac{n.(n-1).(n-2).…. (n-r+1)[(n-r) (n-r-1) (n-r-2)... 3. 2. 1]}{[(n-r) (n-r-1) (n-r-2)....3. 2. 1]}$

P (n, r) = $\dfrac{n!}{(n-r)!}$

Since combinations involve choosing r objects out of n objects where the order doesn't matter, we can determine that:

C(n,r) = the number of permutations /number of ways to arrange ‘r’ objects. $[$Since by the fundamental counting principle, we know that the number of ways to arrange ‘r’ objects in r ways = r!$]$

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

C(n,r) = $\dfrac{\dfrac{n!}{(n-r)!}}{r!}$

Thus we derive C(n,r) =$\dfrac{n!}{r!.(n - r)!}$

Example 1: Find the number of permutations and combinations if the data given is n = 12 and r = 2.

Solution: Given,

n = 12

r = 2

Using the formula given above:

Permutation:

nPr = (n!) / (n-r)! =(12!) / (12-2)! = 12! / 10! = (12 x 11 x 10! )/ 10! = 132

Combination:

$\begin{array}{l}^{n}C_{r} = \frac{n1}{r!(n-r)!}\end{array} $

$\begin{array}{l}\frac{12!}{2!(12-2)!} = \frac{12!}{2!(10)!} = \frac{12\times 11\times 10!}{2!(10)!} = 66\end{array} $

Example 2: Determine how many ways a committee of 5 men and 3 women may be chosen from a group of 9 men and 12 women?

Solution:

Choose 5 men from the group of 9 men = 9C5 ways = 126 ways

Choose 3 women from the group of 12 women = 12C3 ways = 220 ways

The committee was formed in 27720 ways.

Solved Previous year Questions

1. Prove that if each of the ‘m’ points in one straight line is joined to each of the n points on the other straight line, excluding the points on the given two lines. The number of points of intersection of these lines is $\begin{array}{l}\frac{1}{4}\end{array} $ mn(m-1)(n-1).

Solution:

Two points on the first line and two points on the second line are required to obtain one point of intersection. These can be chosen from a set of n points in nC2 ways or from a set of m points in mC2 ways.

Therefore, the required number = mC2 × nC2 = (m(m-1))/2! x (n(n-1))/2! =$\begin{array}{l}\frac{1}{4}\end{array}$

mn(m – 1)(n – 1)

Question 2. Number of divisors of n = 38808 (except 1 and n) is _____.

Solution:

Since, 38808 = 8 × 4851

= 8 × 9 × 539

= 8 × 9 × 7 × 7 × 11

= 23 × 32 × 72 × 11

So, the number of divisors = (3 + 1) (2 + 1) (2 + 1) (1 + 1) = 72.

This includes two divisors 1 and 38808.

Hence, the number of divisors required is = 72 – 2 = 70.

Question 3. A five-digit number divisible by 3 has to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is ________.

Solution:

We know that a five-digit number is divisible by 3 if and only if the sum of its digits (= 15), hence we should avoid using 0 or 3 while creating five-digit numbers.

Now,

(i) So, in the first case, we do not use 0; the five-digit number can be formed (from the digit 1, 2, 3, 4, 5) in 5P5 ways.

(ii) And, in the second case, we do not use 3; the five-digit number can be formed (from the digit 0, 1, 2, 4, 5) in 6 ! −5! × 2 = 480 ways.

The total number of such 5 digit number = 5P5 + (5P5 − 4P4)

= 120 + 96

= 216.

1. In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one woman?

(A) 246

(B) 222

(C) 186

(D) None of these

2. How many numbers greater than 10 lakhs can be formed from 2, 3,0, 3,4,2,3?

(A) 420

(B) 360

(C) 400

(D) 300

**Answer:** 1-(A), 2-(B)

Permutation and combination are explained elaborately in this article, along with the difference between them. We have discussed both the topics here with their formulas, examples and solved questions. With this students will be able when to use permutation and when to use combination formula in a question. Students can also work on Permutations and Combinations of different questions to enhance their knowledge and understanding of this chapter.

See More

JEE Main 2022 June and July Session exam dates and revised schedule have been announced by the NTA. JEE Main 2022 June and July Session will now be conducted on 20-June-2022, and the exam registration closes on 5-Apr-2022. You can check the complete schedule on our site. Furthermore, you can check JEE Main 2022 dates for application, admit card, exam, answer key, result, counselling, etc along with other relevant information.

See More

View All Dates

Application Form

Eligibility Criteria

Reservation Policy

Admit Card

NTA has announced the JEE Main 2022 June session application form release date on the official website https://jeemain.nta.nic.in/. JEE Main 2022 June and July session Application Form is available on the official website for online registration. Besides JEE Main 2022 June and July session application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details. July Session's details will be updated soon by NTA.

It is crucial for the the engineering aspirants to know and download the JEE Main 2022 syllabus PDF for Maths, Physics and Chemistry. Check JEE Main 2022 syllabus here along with the best books and strategies to prepare for the entrance exam. Download the JEE Main 2022 syllabus consolidated as per the latest NTA guidelines from Vedantu for free.

See More

Download full syllabus

Download full syllabus

View JEE Main Syllabus in Detail

JEE Main 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Main Study Materials. It covers the entire JEE Main syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Main study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Main exam 2022 with Vedantu right on time.

See More

All

Mathematics

Physics

Chemistry

See All

Download JEE Main Question Papers & Answer Keys of 2021, 2020, 2019, 2018 and 2017 PDFs. JEE Main Question Paper are provided language-wise along with their answer keys. We also offer JEE Main Sample Question Papers with Answer Keys for Physics, Chemistry and Maths solved by our expert teachers on Vedantu. Downloading the JEE Main Sample Question Papers with solutions will help the engineering aspirants to score high marks in the JEE Main examinations.

See More

In order to prepare for JEE Main 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, RS Aggarwal Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Main 2022 exam so that they can grab the top rank in the all India entrance exam.

See More

See All

JEE Main 2022 free online mock test series for exam preparation are available on the Vedantu website for free download. Practising these mock test papers of Physics, Chemistry and Maths prepared by expert teachers at Vedantu will help you to boost your confidence to face the JEE Main 2022 examination without any worries. The JEE Main test series for Physics, Chemistry and Maths that is based on the latest syllabus of JEE Main and also the Previous Year Question Papers.

See More

NTA is responsible for the release of the JEE Main 2022 June and July Session cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Main 2022 June and July Session is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Main qualifying marks for 2021 ranged from 87.8992241 for general-category, while for OBC/SC/ST categories, they ranged from 68.0234447 for OBC, 46.8825338 for SC and 34.6728999 for ST category.

See More

JEE Main 2022 June and July Session Result - NTA has announced JEE Main result on their website. To download the Scorecard for JEE Main 2022 June and July Session, visit the official website of JEE Main NTA.

See More

Rank List

Counselling

Cutoff

JEE Main 2022 state rank lists will be released by the state counselling committees for admissions to the 85% state quota and to all seats in NITs and CFTIs colleges. JEE Main 2022 state rank lists are based on the marks obtained in entrance exams. Candidates can check the JEE Main 2022 state rank list on the official website or on our site.

Want to know which Engineering colleges in India accept the JEE Main 2022 scores for admission to Engineering? Find the list of Engineering colleges accepting JEE Main scores in India, compiled by Vedantu. There are 1622 Colleges that are accepting JEE Main. Also find more details on Fees, Ranking, Admission, and Placement.

See More

FAQ

**1. What is the contribution of the chapter permutation and combination?**

At least one or two problems from this chapter will appear in JEE Main and other entrance exams every year. You will learn some fundamental counting strategies from the concept of permutation and combinations, which will allow you to answer questions without having to list 3-digit arrangements. In fact, similar methods can be used to determine the number of distinct ways to arrange and pick things without having to list them.

**2. How difficult is the chapter Permutation and Combination?**

You can make an endless number of different types of Permutation and Combination questions, but you still can't be sure you'll be able to answer all of them. So, in order to be able to answer the maximum amount of questions, you must practise a lot. Solve a variety of questions and first try to understand the question. Think about the concept that can be used to solve this question.

**3. What are the real-life examples of permutations and combinations?**

Permutations include arranging individuals, numerals, numbers, alphabets, letters, and colours. Combinations include menu selection, cuisine, clothing, subjects, and the team.

JEE News

JEE Blogs

Trending pages