Exams
Test Series
Previous Year Papers
JEE Main Previous Year Question Paper JEE Advanced Previous Year Papers NEET Previous Year Question Paper CUET Previous Year Papers COMEDK UGET Previous Year Papers UP Polytechnic Previous Year Papers AP POLYCET Previous Year Papers TS POLYCET Previous Year Papers KEAM Previous Year Papers MHT CET Previous Year Papers WB JEE Previous Year Papers GUJCET Previous Year Papers ICAR AIEEA Previous Year Papers CUET PG Previous Year Papers JCECE Previous Year Papers Karnataka PGCET Previous Year Papers NEST Previous Year Papers KCET Previous Year Papers LPUNEST Previous Year Papers AMUEEE Previous Year Papers IISER IAT Previous Year Papers Bihar Diploma DECE-LE Previous Year Papers NPAT Previous Year Papers JMI Entrance Exam Previous Year Papers PGDBA Exam Previous Year Papers AP ECET Previous Year Papers PU CET Previous Year Papers GPAT Previous Year Papers CEED Previous Year Papers AIAPGET Previous Year Papers JKCET Previous Year Papers HPCET Previous Year Papers CG PAT Previous Year Papers SRMJEEE Previous Year Papers BCECE Previous Year Papers AGRICET Previous Year Papers TS PGECET Previous Year Papers MP PAT Previous Year Papers IIT JAM Previous Year Papers CMC Vellore Previous Year Papers ACET Previous Year Papers TS EAMCET Previous Year Papers NATA Previous Year Papers AIIMS MBBS Previous Year Papers BITSAT Previous Year Papers JEXPO Previous Year Papers HITSEEE Previous Year Papers AP EAPCET Previous Year Papers UCEED Previous Year Papers CG PET Previous Year Papers OUAT Previous Year Papers VITEEE Previous Year Papers
Syllabus
JEE Main Syllabus JEE Advanced Syllabus NEET Syllabus CUET Syllabus COMEDK UGET Syllabus UP Polytechnic JEECUP Syllabus AP POLYCET Syllabus TS POLYCET Syllabus KEAM Syllabus MHT CET Syllabus WB JEE Syllabus OJEE Syllabus ICAR AIEEA Syllabus CUET PG Syllabus NID Syllabus JCECE Syllabus Karnataka PGCET Syllabus NEST Syllabus KCET Syllabus UPESEAT EXAM Syllabus LPUNEST Syllabus PUBDET Syllabus AMUEEE Syllabus IISER IAT Syllabus NPAT Syllabus JIPMER Syllabus JMI Entrance Exam Syllabus AAU VET Syllabus PGDBA Exam Syllabus AP ECET Syllabus GCET Syllabus CEPT Syllabus PU CET Syllabus GPAT Syllabus CEED Syllabus AIAPGET Syllabus JKCET Syllabus HPCET Syllabus CG PAT Syllabus BCECE Syllabus AGRICET Syllabus TS PGECET Syllabus BEEE Syllabus MP PAT Syllabus MCAER PG CET Syllabus VITMEE Syllabus IIT JAM Syllabus CMC Vellore Syllabus AIMA UGAT Syllabus AIEED Syllabus ACET Syllabus TS EAMCET Syllabus PGIMER Exam Syllabus NATA Syllabus AFMC Syllabus AIIMS MBBS Syllabus BITSAT Syllabus BVP CET Syllabus JEXPO Syllabus HITSEEE Syllabus AP EAPCET Syllabus GITAM GAT Syllabus UPCATET Syllabus UCEED Syllabus CG PET Syllabus OUAT Syllabus IEMJEE Syllabus VITEEE Syllabus SEED Syllabus MU OET Syllabus
Books
Cut Off
JEE Main Cut Off JEE Advanced Cut Off NEET Cut Off CUET Cut Off COMEDK UGET Cut Off UP Polytechnic JEECUP Cut Off AP POLYCET Cut Off TNEA Cut Off TS POLYCET Cut Off KEAM Cut Off MHT CET Cut Off WB JEE Cut Off ICAR AIEEA Cut Off CUET PG Cut Off NID Cut Off JCECE Cut Off Karnataka PGCET Cut Off NEST Cut Off KCET Cut Off UPESEAT EXAM Cut Off AMUEEE Cut Off IISER IAT Cut Off Bihar Diploma DECE-LE Cut Off JIPMER Cut Off JMI Entrance Exam Cut Off PGDBA Exam Cut Off AP ECET Cut Off GCET Cut Off CEPT Cut Off PU CET Cut Off CEED Cut Off AIAPGET Cut Off JKCET Cut Off HPCET Cut Off CG PAT Cut Off SRMJEEE Cut Off TS PGECET Cut Off BEEE Cut Off MP PAT Cut Off VITMEE Cut Off IIT JAM Cut Off CMC Vellore Cut Off ACET Cut Off TS EAMCET Cut Off PGIMER Exam Cut Off NATA Cut Off AFMC Cut Off AIIMS MBBS Cut Off BITSAT Cut Off BVP CET Cut Off JEXPO Cut Off HITSEEE Cut Off AP EAPCET Cut Off GITAM GAT Cut Off UCEED Cut Off CG PET Cut Off OUAT Cut Off VITEEE Cut Off MU OET Cut Off
Latest Updates
Eligibility
JEE Main Eligibility JEE Advanced Eligibility NEET Eligibility CUET Eligibility COMEDK UGET Eligibility UP Polytechnic JEECUP Eligibility TNEA Eligibility TS POLYCET Eligibility KEAM Eligibility MHT CET Eligibility WB JEE Eligibility OJEE Eligibility ICAR AIEEA Eligibility CUET PG Eligibility NID Eligibility JCECE Eligibility Karnataka PGCET Eligibility NEST Eligibility KCET Eligibility LPUNEST Eligibility PUBDET Eligibility AMUEEE Eligibility IISER IAT Eligibility Bihar Diploma DECE-LE Eligibility NPAT Eligibility JIPMER Eligibility JMI Entrance Exam Eligibility AAU VET Eligibility PGDBA Exam Eligibility AP ECET Eligibility GCET Eligibility CEPT Eligibility PU CET Eligibility GPAT Eligibility CEED Eligibility AIAPGET Eligibility JKCET Eligibility HPCET Eligibility CG PAT Eligibility SRMJEEE Eligibility BCECE Eligibility AGRICET Eligibility TS PGECET Eligibility MP PAT Eligibility MCAER PG CET Eligibility VITMEE Eligibility IIT JAM Eligibility CMC Vellore Eligibility AIMA UGAT Eligibility AIEED Eligibility ACET Eligibility PGIMER Exam Eligibility CENTAC Eligibility NATA Eligibility AFMC Eligibility AIIMS MBBS Eligibility BITSAT Eligibility JEXPO Eligibility HITSEEE Eligibility AP EAPCET Eligibility GITAM GAT Eligibility UPCATET Eligibility UCEED Eligibility CG PET Eligibility OUAT Eligibility IEMJEE Eligibility SEED Eligibility MU OET Eligibility

Permutation and Combination Definition, Formulas & Examples

Last Updated on Jul 09, 2025
Download As PDF
IMPORTANT LINKS

Permutations and combinations are two basic ideas in mathematics that help us count and arrange things. They are part of a branch of math called combinatorics, which is all about figuring out how many ways we can choose or arrange items from a group.

A permutation is used when the order of items matters. For example, if you're arranging people in a line or assigning positions like first, second, and third, the order changes the result, so we use permutations.

Maths Notes Free PDFs

Topic PDF Link
Class 12 Maths Important Topics Free Notes PDF Download PDF
Class 10, 11 Mathematics Study Notes Download PDF
Most Asked Maths Questions in Exams Download PDF
Increasing and Decreasing Function in Maths Download PDF

A combination is used when the order doesn’t matter. For example, if you're choosing toppings for a pizza or selecting team members, it doesn't matter in which order you pick them, so we use combinations.

These concepts are used in many real-life situations like creating passwords, planning events, choosing teams, and even solving puzzles. They are also important in statistics, probability, and computer science.

Table of Contents:

This article aims to provide a comprehensive understanding of permutations and combinations. It covers their definitions, formulas, differences, uses, and some solved examples. It also includes a Permutation And Combination Worksheet for students to practice and enhance their understanding of these concepts.

What is a Permutation?

In mathematics, a permutation means arranging the items of a group in a specific order. When you change the order of things, you are making a new permutation. For example, if you have three letters A, B, and C, then ABC, BCA, and CAB are all different permutations because the order is different in each case. Permutations are useful when the position or sequence of items matters. This concept is often used in problems involving arranging people, numbers, or objects, and is important in subjects like probability, algebra, and logic.

Learn more about Permutation here.

UGC NET/SET Course Online by SuperTeachers: Complete Study Material, Live Classes & More

Get UGC NET/SET SuperCoaching @ just

₹25999 ₹11666

Your Total Savings ₹14333
Explore SuperCoaching

What is a Combination?

A combination is a way of choosing items from a group where the order doesn't matter. For example, choosing two fruits from apple, banana, and orange — picking apple and banana is the same as picking banana and apple. In combinations, we are only interested in which items are chosen, not how they are arranged. When we select k items out of n without repeating any, it’s called a simple combination. If items can be repeated, it’s called a combination with repetition or k-selection. Combinations are often used in probability, statistics, and everyday choices.

Get more insights about Combination here.

Formulas for Permutation and Combination

There are several formulas associated with the concepts of permutation and combination. The two fundamental formulas are:

Test Series
133.7k Students
NCERT XI-XII Physics Foundation Pack Mock Test
323 TOTAL TESTS | 3 Free Tests
  • 3 Live Test
  • 163 Class XI Chapter Tests
  • 157 Class XII Chapter Tests

Get Started

Permutation Formula

A permutation involves the selection of 'r' items from a set of 'n' items, where the order of selection matters and replacement is not allowed.

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

Combination Formula

A combination involves the selection of 'r' items from a set of 'n' items, where the order of selection doesn't matter and replacement is not allowed.

Permutation and Combination Made Simple

Permutation and Combination are important math concepts that help us count the number of ways things can be arranged or selected.

A permutation is used when the order matters. For example, if you are arranging 3 books on a shelf, placing them in different orders counts as different permutations. The formula to find the number of permutations is:
nPr = n! / (n – r)!

A combination is used when the order doesn’t matter. For example, if you are choosing 2 friends from a group of 5 to go on a trip, the order you choose them doesn’t matter. The formula for combinations is:
nCr = n! / [r! × (n – r)!]

These topics are used in probability, statistics, and everyday decisions, like forming teams, passwords, or schedules. In exams like SSC, Banking, and Railways, questions on this topic test your logical and analytical thinking.

To master this topic, it's important to understand whether order is important in the given problem. Once that’s clear, use the right formula and solve with confidence!.

Derivation of Permutation and Combination Formulas 

We can understand how the formulas for permutation and combination are formed by using basic counting methods. Let’s break it down simply.

Derivation of Permutation Formula:

Permutation means choosing and arranging r items out of n total items, where order matters.

Let’s say you want to pick r items from n, one after another:

  • First choice: n options
  • Second choice: (n - 1) options
  • Third choice: (n - 2) options
  • ... and so on, until r items are chosen.

So, the total number of ways to arrange r items out of n is:

P(n, r) = n × (n − 1) × (n − 2) × ... up to r terms

Now, to write this in a simpler form, we multiply and divide by (n − r)!:

P(n, r) = n! / (n − r)!

Derivation of Combination Formula:

Combination means choosing r items from n items, but order does not matter.

From above, we know:

P(n, r) = n! / (n − r)!

But in combinations, all different orders of the same items are considered the same, so we divide by r! (the number of ways to arrange r items):

C(n, r) = P(n, r) / r! = n! / [(n − r)! × r!]

So, the final formula for combinations is:

C(n, r) = n! / [(n − r)! × r!]

Difference Between Permutation and Combination

Check out the differences between permutation and combination below.

Permutation

Combination

Arranging digits, letters, and colors

Selecting a menu, clothes, subjects, team

Choosing a team leader and team members from a group

Selecting team members from a group

Picking two favorite colors in order from a color palette

Picking two colors from a color palette

Picking first, second, and third place winners

Picking three winners

Applications of Permutation and Combination

Permutation and Combination are powerful mathematical tools used to count, arrange, or select objects in a variety of ways. These concepts are widely used in daily life, academics, and different fields such as business, technology, and science.

1. Arrangements (Permutations)

Permutations are used when we want to arrange objects and the order of arrangement matters. This is important in cases such as:

  • Seating people in a row or around a table
  • Arranging books on a shelf
  • Assigning ranks or positions in a competition
  • Generating passwords or ATM PINs (where the sequence matters)

2. Selections (Combinations)

Combinations are used when we want to select items or people, but the order of selection does not matter. Common examples include:

  • Choosing members for a team or committee
  • Selecting lottery numbers
  • Picking items from a menu
  • Choosing clothes to wear (shirt + trousers, regardless of order)

3. Decision-Making

In real-life problems, permutation and combination help us figure out the number of possible choices or outcomes. For example:

  • How many different outfits can be made from 3 shirts and 2 trousers?
  • In how many ways can we form a 3-member team from 10 people?
  • How many different ways can top 3 positions be awarded from 15 participants?

4. Business and Technology

These concepts are used in:

  • Computer programming (algorithm optimization, coding theory)
  • Cryptography (securing data through code combinations)
  • Network design and telecommunication (assigning IP addresses or channels)
  • Scheduling tasks or events efficiently

5. Science and Research

In biology and chemistry:

  • Gene arrangements and DNA sequencing
  • Different combinations of compounds or molecules
  • Possible outcomes in experiments

Solved Examples on Permutations and Combinations

Example 1:

Determine the number of permutations and combinations for n = 10 and r = 3.

Solution:

Given n = 10 and r = 3, we can use the permutation and combination formulas as follows:

Permutation:

ⁿPᵣ = (n!) / (n - r)! = (10!) / (10 - 3)! = 720

Combination:

ⁿCᵣ = n! / [r!(n - r)!] = 10! / [3!(10 - 3)!] = 120

Example 2: If the letters of the word "HELLO" are arranged in all possible ways and listed in dictionary order, what is the 49th word?

Solution:

The word "HELLO" has the letters: H, E, L, L, O

Step 1: Arrange the letters in alphabetical order:
E, H, L, L, O

Now we find out how many words start with each letter (like a dictionary):

Start with E:
Remaining letters: H, L, L, O
Total arrangements = 4! / 2! = 12

Words 1 to 12

Start with H:
Remaining letters: E, L, L, O
Total arrangements = 4! / 2! = 12

Words 13 to 24

Start with L:
Remaining letters: E, H, L, O
Total arrangements = 4! = 24
(Because only one L is left to arrange, no repetition)

Words 25 to 48

Now we have listed 48 words so far.
The 49th word will be the first word starting with O.

Remaining letters: E, H, L, L
Arranged in dictionary order: E < H < L < L

So the first word formed using O + E, H, L, L is:

49th word = OEHLL

OEHLL is the 49th word in dictionary order.

Example 3:

In how many ways can a committee of 5 men and 3 women be selected from 8 men and 10 women?

Solution:

Selecting 5 men out of 8 = 8C5 ways = 56 ways

Selecting 3 women out of 10 = 10C3 ways = 120 ways

Total number of ways = (56 x 120) = 6720 ways. So, the committee can be selected in 6720 ways.

Practice Questions on Permutations and Combinations

Question 1: How many ways can the letters of the word "MATH" be arranged so that all the vowels come together?

Question 2: How many teams of 4 girls and 3 boys can be formed from a group of 8 girls and 9 boys?

Question 3: How many words can be formed by 3 vowels and 4 consonants taken from 5 vowels and 7 consonants?

Important Links
NEET Exam
NEET Previous Year Question Papers NEET Mock Test NEET Syllabus
CUET Exam
CUET Previous Year Question Papers CUET Mock Test CUET Syllabus
JEE Main Exam
JEE Main Previous Year Question Papers JEE Main Mock Test JEE Main Syllabus
JEE Advanced Exam
JEE Advanced Previous Year Question Papers JEE Advanced Mock Test JEE Advanced Syllabus

More Articles for Maths

Frequently Asked Questions For Permutation and Combination

Permutation is the arrangement of a set of objects in a specific order. The order of objects is important in permutations. It is used when we are arranging people, digits, letters, etc., and every different order is considered a unique permutation. For example, the permutations of A, B, and C taken 2 at a time are: AB, BA, AC, CA, BC, CB.

Combination is the selection of items from a group, where the order of selection does not matter. It is used when we are only interested in which items are chosen, not how they are arranged. For example, the combinations of A, B, and C taken 2 at a time are: AB, AC, BC (AB and BA are the same combination).

Factorial means the product of all positive integers up to n. Example: 5!=5×4×3×2×1=1205! = 5

Permutation is the arrangement of items where order matters. Combination is the selection of items where order does not matter.

Permutations and combinations are used in real life for decision-making, probability, cryptography, event arrangements, seating plans, lottery systems, and calculating possible outcomes. They are essential in fields like mathematics, statistics, computer science, and operations research.

Use permutation when order matters — like arranging people in a line or creating passwords.

Use combination when order does not matter — like choosing a team from a group.

Report An Error