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1330 LearnersLast updated on October 6, 2025
Permutations
Permutations refer to different ways of arranging objects in a specific order. It is the rearrangement of a set of items in a specific linear order. The symbol nPr is used to indicate the number of permutations of n distinct objects, taken r at a time. In this topic, let's learn about permutations in detail.
What are Permutations?
The number of ways a set of objects can be arranged is known as permutations. For instance, if there are 5 books and 3 rows on a shelf, the number of ways the books can be arranged is calculated using permutations. Permutations can be classified into different types. They are permutations with repetition, without repetition, multi-sets, and circular permutations. The key takeaways of permutations are listed below:
- It refers to the number of possible ways to arrange a set of items or objects.
- The order of numbers is important when using a permutation.
- The two common types of permutations are with repetition and without repetition. Permutations with multi-sets and circular permutations are less prevalent.
- From a single combination, several permutations are possible.
- Permutations and combinations are different. Combinations are selections of data from a group where the order does not matter.
Next, let us explore how to calculate permutations in detail.
How to Calculate Permutations?
Here, let’s see how we can find permutations. The general formula we use to find permutations is:
P(n, r) = n! / (n − r)!
Here, n is the total number of elements in the data set.
r is the total number of selected elements in a specific order.
! is the factorial.
For instance, let's say we have 10 different books and want to select and arrange 2 of them. We can calculate the number of ways to do it using permutations.
Without repetition:
The formula is: nPr = 10! / (10 − 2)! = 10! / 8! = (10 × 9 × 8!) / 8! = 90
So, there are 90 unique ways to arrange 2 books from a set of 10.
With repetition, the formula is:
10\(^2\) = 10 × 10 = 100
Hence, there are 100 ways (with repetition) to arrange 2 books from a set of 10.
Another key concept is factorials, and they are useful in permutations.
For example, the factorial of 8 = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320.
Difference Between Permutations and Combinations
Permutations and combinations are methods used to determine the number of possible arrangements of elements. So, let's see how they differ from each other.
| Permutation | Combination |
| In permutation, the order of the data is considered. | In combinations, the order of data is not considered. |
| In permutations, elements are selected from a list. | In combination, the data is chosen from a group. |
| The data is specifically arranged. | Here there is a selection of data. |
Real-Life Applications of Permutations
Now let’s learn how we use permutations in the real world.
- In cybersecurity and encryption, we use permutations to create passwords to secure our systems.
- In lottery games, we use permutations to pick from a pool of numbers.
- Permutations can be applied to various scenarios, such as arranging people, and seats, forming teams, and so on.
- In DNA and genetics, permutations help in analyzing sequences of a DNA.
Common Mistakes and How to Avoid Them in Permutations
Students tend to make mistakes when learning inter-related concepts like permutations and combinations. So let’s check out some common mistakes and ways to avoid them when learning permutation.
Mistake 1
Confusion between permutations and combinations
Confusion between permutations and combinations is common among students, as both look similar. To avoid this error, they need to understand the difference between them. Permutation refers to the arrangement of objects where order matters, whereas combinations involve the selection of objects where order doesn't matter.
Mistake 2
Counting the duplicate arrangements
When the objects are similar, counting each instance will lead to unique outcomes. So the items are divided by the factorial of the number of times each object is repeated, that is, n! / r!
Mistake 3
Not arranging the data in sequence
Failing to arrange the data in sequence can lead to errors due to miscounting or overlooking certain conditions. So students should understand the problem and break it down to minimize errors.
Mistake 4
Not reducing the factorial terms
Students sometimes make mistakes by using full factorials instead of simplifying them.
For instance, when calculating n! and (n − k)!, they might expand both factorials unnecessarily. Instead, they can cancel out the common factorials and simplify the equation efficiently.
Mistake 5
Calculation errors
Calculation errors are common among students when working with factorials or large numbers. To avoid it, students should break it down step by step and do the basic operations correctly. Try to verify whether the final answer is correct or not.
Solved Examples of Permutations
Problem 1
In how many ways can 5 different books be arranged on a shelf?
120
Explanation
To find the permutations, we use the formula n!
Here, n = 5
So n! = 5! = 5 × 4 × 3 × 2 × 1 = 120
So, we can arrange the books in 120 ways.
Problem 2
How many 3-letter words (with distinct letters) can be formed using the letters A, B, C, D, and E?
Here, we can arrange the given letters into 60 different three-letter words.
Explanation
To find the number of possibilities of 3-letter arrangements, we find the permutation using the formula:
p(n, k) = n! / (n − k)!
Here, n = 5 and k = 3
P(5, 3) = 5! / (5 − 3)! = 5! / 2!
= (5 × 4 × 3 × 2 × 1) / (2 × 1)
Next, cancel out the common terms (2 × 1):
So, 5 × 4 × 3 = 60
Problem 3
In how many ways can the letters in the word 'GOLD' be arranged?
The word 'gold' can be arranged in 24 different ways.
Explanation
The word 'GOLD' can be arranged in n! ways.
Here, n = 4 (the word 'GOLD' consists of 4 letters).
So we start with 4 and then multiply 4 by the next smallest number, 3.
4 × 3 = 12
Again, multiply the result by 2:
12 × 2 = 24
Finally, multiply 24 by 1.
24 × 1 = 24.
So, n! = 4! = 4 × 3 × 2 × 1 = 24
So, the word 'GOLD' can be arranged in 24 different ways.
Problem 4
In how many ways can 8 athletes be assigned 3 distinct positions in a relay race?
There are 336 ways for 8 athletes to try 3 distinct positions in a relay race.
Explanation
We use the permutation formula,
p(n, k) = n! / (n − k)!
Here, n = 8 and k = 3
So, p(8, 3) = 8! / (8 − 3)! = 8! / 5!
= 8 × 7 × 6 × 5! / 5!
= 8 × 7 × 6 = 336
Problem 5
A locker password consists of 5 distinct digits chosen from 1 to 9. How many passwords can be created?
15,120 passwords can be created.
Explanation
We use the permutation formula,
p(n, k) = n! / (n − k)!
Here, n = 9 and k = 5
So, p(9, 5) = 9! / (9 − 5)! = 9! / 4!
= 9 × 8 × 7 × 6 × 5 × 4! / 4!
= 9 × 8 × 7 × 6 × 5 = 15,120.
FAQs on Permutations
1.What is a permutation?
2.What is the formula for permutation?
3.What is the difference between a permutation and a combination?
4.What are some real-life applications of permutations?
5. What are the types of permutations?
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Jaipreet Kour Wazir
About the Author
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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