1225 LearnersLast updated on June 5th, 2025
Permutations
Permutations refer to different ways of arranging objects in a specific order. It is also referred to as the rearrangement of the already ordered set of items in a 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 books can be arranged is calculated using the 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. Though 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 order does not matter.
Next, let us explore how to calculate permutations in detail.
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How to calculate Permutations
As we discussed permutations, now let’s see how we 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
k is the total number of selected elements in a specific order.
! is the factorial.
For instance, if we have 10 different books and want to select and arrange 2 of them. Now we can calculate the number of ways using permutations.
Without permutations:
The formula is: nPr = 10! / (10 - 2)! = 10! / 8! = 90
So, there are 90 possible ways to arrange 2 books from a set of 10.
With repetition:
The formula is:
102 = 10 × 10 = 100
Hence, there are 100 different ways 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 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
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 permutation, the data is chosen 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 our real-life applications.
- In cybersecurity and encryption, we use permutations to create passwords to secure our systems.
- In lottery games, we use permutations to pick a pool of numbers
- Permutations can apply to various scenarios, such as arranging people, and seats, forming teams, and so on.
- In DNA and genetics, we use it to arrange them in sequence of the DNA
Common Mistakes and How to Avoid Them in Permutations
Students tend to make mistakes when learning related concepts like permutations and combinations. So let’s check out some common mistakes and ways to avoid them when learning permutation.
Mistake 1
Confusing with permutations and combinations
Confusing 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 sequence can lead to errors due to miscounting or overlooking conditions. So students should understand the problem and break it down the problem by visualizing it to minimize errors.
Mistake 4
Not reducing the factorial terms
Students sometimes make errors 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 down step by step and do the basic operations. Try to verify whether the final answer is correct or not.
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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 letters into 60 different words
Explanation
To find the numbers 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):
= 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! Time
Here, n= 4 (the word GOLD consists of 4 letters).
Here, we start with 4. Then we multiply 4 by the next smallest number, 3.
4 × 3 = 12
Again, multiply the result by 2:
12 × 2 = 24
Finally, multiply the 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?
The possible ways to assign the positions are 336 ways
Explanation
we use the permutation format,
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?
The possible ways to create a password is 15,120
Explanation
we use the permutation format,
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 = 15120.
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FAQs on Permutations
1.What is a permutation?
2.What is the formula for Permutations?
3.What is the difference between a permutation and a combination?
4.What are real-life applications of Permutations?
5. What are the types of permutations?
6.How can children in Canada use numbers in everyday life to understand Permutations?
7.What are some fun ways kids in Canada can practice Permutations with numbers?
8.What role do numbers and Permutations play in helping children in Canada develop problem-solving skills?
9.How can families in Canada create number-rich environments to improve Permutations skills?
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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
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
