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215 LearnersLast updated on October 22, 2025
Difference Between Permutations and Combinations
Permutations and combinations are methods used to arrange or select items from a larger set. The key difference lies in whether the order of selection matters.
What is Permutation?
A permutation is the number of ways to arrange a set of items in a specific order.
It is represented as \(^nP_r\), where n is the total number of items and r is the number of items chosen for the arrangement.
The formula to calculate Permutation is:
\(^nP_r = \frac {n!} {(n – r)!}\)
What is Combination?
A combination is the selection of items from a larger set where the order of selection is not important.
It is represented by \(^nC_r\), where n is the total number of distinct items and r is the number of items selected.
The formula for combination is:
\(^nC_r = \frac {n!} {[r! × (n – r)!]}\)
Difference Between Permutations and Combinations
The order of selection of items is the main difference between permutations and combinations. Now we will learn the difference between permutations and combinations in detail.
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Permutation |
Combination |
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A permutation is the number of ways of arranging items in a specific order. |
A combination is the total number of possible selections of items from a given set. |
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In permutation, the order plays an important role. |
The order of the items is not important for the combination. |
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When the items are of different kinds, we use permutations. |
When the items are of a similar kind and when order does not matter, we use combinations. |
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The possible outcomes of tossing two coins are: {HH, HT, TH, TT}. Here, HT and TH are different, as in order matters. |
When tossing two coins, the possible outcomes are: {HH, HT, TT}. As order does not matter in combinations, we consider HT and TH as one. |
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Permutation formula: \(^nP_r = \frac {n!} {(n – r)!}\) |
Combination formula: \(^nC_r = \frac {n!} {[r! × (n – r)!]}\) |
What is the Relation Between Permutation and Combination?
By comparing the formulas of permutation and combination, let’s understand the relationship between permutation and combination.
When selecting and arranging r items from n, we can express the permutation as the product of r! And the combination of those r items.
\(^nP_r = \frac {n!} {(n – r)!}\)
We can rewrite it as:
\(^nP_r = \frac {(r! × n!)}{[r! × (n – r)! ]}\)
\(^nP_r = r! ×\ ^nC_r \)
Thus, the number of permutations is the product of the number of combinations and the ways to arrange the selected items (r!).
Tips and Tricks to Master Difference Between Permutations and Combinations
Here are some fun, easy-to-remember tips and tricks to help students and parents master the difference between permutations and combinations
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Remember the golden rule that, while finding the permutation, position matters. On the other hand, to find the combination, position doesn’t matter.
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You can help yourself before solving a problem by always asking if the order makes a difference. If they do, then it is a permutation problem and if it does not, then we have to find the combinations.
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The easiest way to identify these formulas is to remember that, if the formula includes r! in the denominator, it’s a combination, because it removes duplicate orders.
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If you are confused about permutations and combinations in the middle of a question, then remember that if we are arranging people, numbers, or letters, we must use permutation. On the other hand, if we are choosing people, numbers, or letters, then we must use combination.
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To make your calculation faster while dealing with permutations, try counting ways to seat family members at dinner. For combinations, count ways to select friends for a group photo.
Common Mistakes and How to Avoid Them in Difference Between Permutations and Combinations
Students often confuse permutations and combinations, leading to errors. So we will be learning some common mistakes and ways to avoid them.
Mistake 1
Confusing permutations and combinations as the same
Students think that permutations and combinations are the same, which leads to errors. So always remember that in permutations the order of selection matters, and in combinations the order of selection does not matter.
Mistake 2
Using the wrong formula to find the permutations and combinations
One common mistake students make when finding the permutations and combinations is applying the wrong formula, which can lead to errors. So always memorize the formulas: Permutation formula: \(^nP_r = \frac {n!} {(n – r)!}\) and combination formula: \(^nC_r = \frac {n!} {[r! × (n – r)!]}\)
Mistake 3
Unable to identify when repetition is allowed
Sometimes students find it hard to determine whether the items can be repeated or not, which can result in using incorrect formulas. So always read the problem carefully and identify the phrases like “repeating allowed”, “no digit repeated”, or “no repeats”.
Mistake 4
Ignoring step-by-step logic
Students often memorize formulas but don’t understand why they work. As a result, they can’t decide which one to apply in complex problems. Encourage students to write the logic before using formulas:
“We are arranging objects, so order matters → Permutation.”
“We are choosing a group, so order doesn’t matter → Combination.”
Mistake 5
Confusing permutations with combinations in word problems
When doing problems related to permutations or combinations, students fail to identify whether it is permutations or combinations. To avoid this confusion, it is important to read the questions carefully. Identify the words like arrange, order, or sequence for permutation and words like select, choose, or group for combination.
Real-World Applications of Difference Between Permutations and Combinations
In real life, we use permutations and combinations from creating passwords to scheduling events. In this section, we will learn a few applications of the difference between permutations and combinations.
- We use permutations to generate passwords, as the order of characters matters in passwords. The order of digits or letters changes the password completely.
- We use permutations in cybersecurity and cryptography to create unique codes and passwords, as the order of characters is important.
- When you pick members for a team, it doesn’t matter in which order they are selected. Because selection matters, not order, it’s a combination. For example, forming school committees or project groups
- When choosing pizza toppings, the order doesn’t matter; cheese, mushroom, and corn is the same as corn, cheese, and mushroom. Since the combination is what matters, not order, this is a combination.
- The permutations and combinations are used to assign phone numbers, city codes, car numbers, and telephone codes, as the order of the characters is important.
Solved Examples of Difference Between Permutations and Combinations
Problem 1
From a set of 5 different books, in how many ways can you select 3 books and arrange them on a shelf?
The number of ways to choose 3 books is 10.
The ways of arranging 3 books from a set of 5 are 60.
Explanation
The number of ways to select books: \(^nC_r = \frac {n!} {[r! × (n – r)!]}\)
Here, n = 5 and r = 3
\(^5C_3 = \frac {5!} {[3! × (5 – 3)!]}\\ ^5C_3= \frac {5!} {[3! × 2!]}\\ ^5C_3= \frac {(5× 4× 3! )}{(3! ×2!)}\\ ^5C_3= \frac {(5× 4 )}{2!}\\ ^5C_3= \frac {5×4}{2×1}\\ ^5C_3=\frac {20}{2}\\ ^5C_3= 10\)
To find the ways of arranging 3 books on a shelf, we calculate the permutations:
\(^nP_r = \frac {n!} {(n – r)!}\)
Here, n = 5 and r = 3
\(^nP_r =\frac {5!} {(5 – 3)!}\\ ^5P_3= \frac {5!} {2!} \\ ^5P_3= \frac {(5 × 4 × 3 × 2!)} { 2!}\\ ^5P_3= 5 × 4 × 3 \\ ^5P_3= 60\)
Problem 2
In how many ways can you arrange 4 out of 7 different letters?
We can arrange the letters in 840 ways.
Explanation
To find the ways of arranging 4 out of 7 different letters, we use the permutation formula.
\(^nP_r = \frac {n!} {(n – r)!}\)
Here, n = 7 and r = 4
\(nPr = \frac {7!}{(7 – 4)!}\\ ^7P_4 = \frac {7!}{3!} \\ ^7P_4= \frac {(7 × 6 × 5 × 4 × 3!)}{3!}\\ ^7P_4= 7 × 6 × 5 × 4 \\ ^7P_4= 840\)
Problem 3
How many ways can you form a team of 4 from 10 players?
The number of ways you can form a team of 4 from 10 players is 210.
Explanation
To find how many ways we can form a team of 4 people from a group of 10, we use the combination formula
\(^nC_r = \frac {n!} {[r! × (n – r)!]}\)
Where n = 10 and r = 4
\(C(10, 4) = \frac {10! } {[4! × (10 – 4)!]}\\ ^{10}C_4 = \frac {10!}{(4! × 6!)}\\ ^{10}C_4= \frac {(10 × 9 × 8 × 7 × 6!)}{(4! × 6!)}\\ ^{10}C_4= \frac {(10 × 9 × 8 × 7)} {(4 × 3 × 2 × 1)}\\ ^{10}C_4= \frac {5040}{24}\\ ^{10}C_4= 210\)
Problem 4
In how many ways can 5 people be arranged in a line from a group of 9?
We can arrange them in 15120 ways
Explanation
To find the ways to arrange 5 people in a line from a group of 9, we use the permutation formula
\(^nP_r = \frac {n!} {(n – r)!}\)
Here, n = 9 and r = 5
\(^9P_5 = \frac {9!} {(9 – 5)!}\\ ^9P_5= \frac {9!} {4!}\\ ^9P_5= \frac {(9 × 8 × 7 × 6 × 5 × 4!)} {4!}\\ ^9P_5= 15120\)
Problem 5
Out of 7 runners, how many ways can you select 3 to join a relay team?
In 35 ways, we can select the relay team
Explanation
To find the number of ways to select 3 runners to join a relay team, we find the combination
\(^nC_r = \frac {n!} {[r! × (n – r)!]}\)
Here, n = 7 and r = 3
\(C(7,3) = \frac {7! }{ [3! × (7 – 3)!]}\\ ^7C_3= \frac {7!}{[3! × (7 – 3)!]}\\ ^7C_3= \frac {7! }{(3! × 4!)}\\ ^7C_3= \frac {(7 × 6 × 5 × 4!)}{(4! × 3 × 2 × 1)}\\ ^7C_3= \frac {(7 × 6 × 5)}{(3 × 2 × 1)}\\ ^7C_3= 35\)
FAQs on Difference Between Permutations and Combinations
1.What is the main difference between permutations and combinations?
2.What is the formula for a combination?
3.What is the formula for permutations?
4.How many combinations of 2 out of 5?
5.What is 2!?
6.What is the easiest way to explain permutations and combinations to my child?
7.How can I help my child remember which is which?
8.Why is it important for my kid to learn this concept?
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Jaipreet Kour Wazir
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Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
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