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Permutation
By Karan Singh Bisht
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Updated on 29 Apr 2025, 11:58 IST
In mathematics, permutation refers to the arrangement of objects in a specific order. Unlike combinations, where the order does not matter, permutations focus on different ways objects can be arranged.
Example: With three letters A, B, and C, the permutations are: ABC, ACB, BAC, BCA, CAB, and CBA.
Permutations are used in probability, computer science, cryptography, seating arrangements, passwords, and scheduling.
What is Permutation?
Permutation refers to the arrangement of objects in a specific order. When working with permutations, both the selection and the sequence of arrangement are important. In essence, the order in which items are arranged plays a crucial role. Simply put, a permutation is an ordered combination.
Permutation Formula
The general formula is:
P(n, r) = n! / (n − r)!
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- n = total number of objects
- r = number selected
- ! = factorial
Example: Arranging 3 books out of 5:
P(5,3) = 5! / 2! = 60
Types of Permutations
1. Without Repetition
Use standard formula: P(n, r) = n! / (n − r)!
Example: 4-digit PIN using 1-9: P(9,4) = 3024
2. With Repetition
Use: n! / (n1! × n2! × ...) where n1, n2... are repeating item counts.
Example: LEVEL → 5! / (2! × 2!) = 30
Difference Between Permutations and Combinations
| Feature | Permutation | Combination |
| Order Matters | Yes | No |
| Formula | n! / (n - r)! | n! / (r!(n - r)!) |
| Example | Arrange 5 books in 3 slots | Choose 3 books out of 5 |
| Used In | PINs, seatings | Lottery, team selection |
Real-Life Applications
- Seating Arrangements: P(5,5) = 120
- Password Generation: P(26,4) = 358,800
- Lottery System: P(49,6) = 49! / 43!
| Also Check |
| Associative Property |
| Centroid of a Triangle |
| Collinear Points |
| Commutative Property |
| Cos 0 |
| Hypotenuse |
Conclusion
Permutation helps solve real-life problems like generating passwords, arranging seats, and scheduling. It's essential in competitive exams, computer science, and probability studies.
Permutation FAQs
What is a permutation and example?
A permutation is an arrangement of objects in a specific order. Unlike combinations, the order of elements in permutations matters.
Example: The permutations of the letters A, B, and C include: ABC, ACB, BAC, BCA, CAB, and CBA. Each unique arrangement is counted as a different permutation.
What are the permutations of 1, 2, 3, 4?
The permutations of the digits 1, 2, 3, 4 are all the possible ways to arrange them in different orders. Since there are 4 digits, the number of permutations is:
4! = 4 × 3 × 2 × 1 = 24
Here are some examples of the 24 permutations:
- 1234
- 1243
- 1324
- 1342
- 1432
- 4321
How many 3-digit numbers can be formed using 1-9?
P(9,3) = 504
What is the permutation of 5?
The permutation of 5 refers to the total number of ways to arrange 5 unique elements. This is calculated using factorial notation:
5! = 5 × 4 × 3 × 2 × 1 = 120
So, there are 120 permutations of 5 distinct items.
What is 7 permutation 2?
The expression 7 permutation 2 (written as P(7, 2)) means the number of ways to arrange 2 items out of 7 in order.
P(7, 2) = 7! / (7 - 2)! = 7 × 6 = 42
So, there are 42 possible permutations when choosing and arranging 2 out of 7 items.
What are the 3 types of permutation?
The three main types of permutations are:
- Permutation without Repetition – No element is repeated. Example: Arranging 3 out of 5 different books.
- Permutation with Repetition – Elements can repeat. Example: Creating a 4-digit PIN using digits 0–9.
- Circular Permutation – Arrangements are made in a circle, like seating people around a table. Example: 5 people sitting around a round table has (5 - 1)! = 24 circular permutations.