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.

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:

• Definition of Permutation
• Definition of Combination
• Formulas
• Difference Between Permutation and Combination
• Uses
• Solved Examples
• Practice Questions

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.

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:

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.

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?