Permutations help us understand how many different ordered arrangements can be formed from a set of objects. When the order matters, we use permutations. Whether it's arranging students, creating passwords, or ranking players, permutations offer a clear mathematical way to count all possible arrangements. This makes the formula essential for school maths and competitive exams.
Permutation Formula Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| nPr = n! / (n – r)! | n = total objects, r = selected objects, ! = factorial | Used when order matters (seating, passwords, rankings) |
What is Permutation in Maths?
A permutation in mathematics means arranging objects in a specific order. This is different from combinations because in permutations, the position of each object changes the outcome. For example, arranging students A, B, C in order is different from C, B, and A, even though the objects are the same.
The formula nPr = n! / (n – r)! calculates how many ordered arrangements you can make when selecting r objects from n total objects.
Here, n! (n factorial) means multiplying all natural numbers from n down to 1.
Dividing by (n – r)! removes the parts that don’t involve the selected objects.
This formula is widely used in creating passwords, seating arrangements, generating security codes, and ranking participants. It's also very important for competitive exams like CUET, SSC, Banking, and JEE.
Examples to Calculate Permutations
Example 1: Seating 3 students in 5 chairs
Step 1: Use the formula → nPr = n! / (n – r)!
Step 2: n = 5, r = 3
Step 3: 5P3 = 5! / (5 – 3)! = 120 / 2 = 60
So, 60 different ways are possible.
Example 2: Arranging 3 letters from the word MATH
Step 1: Total letters (n) = 4
Letters to arrange (r) = 3
Step 2: Apply formula →
4P3 = 4! / (4 – 3)!
Step 3: Solve
4! = 24
(4 – 3)! = 1! = 1
So,
4P3 = 24 / 1 = 24
Thus, 24 different arrangements of 3 letters can be made from “MATH.”
FAQs about Permutation Formula
Q1. What is the difference between a permutation and a combination?
Permutation is about arrangement (order matters), while combination is about selection (order doesn’t matter).
Q2. Can nPr be zero?
Yes. If r > n, then nPr = 0.
Q3. What does 0! equal?
0! = 1 by definition.
Q4. Where do we use permutations in real life?
In creating passwords, seat arrangements, ranking, scheduling, and coding systems.
Q5. Is permutation important for competitive exams?
Yes, it is frequently asked in CUET, SSC, Banking, JEE, and many aptitude-based exams.