Permutation without Repetition Calculator
Calculate permutations nPr where order matters and elements cannot be repeated
Calculate Permutations without Repetition
Total number of distinct objects available
Number of objects to arrange in order
Number of Permutations
Formula
P(n,r) = n! / (n-r)!
Where n = 10 (total objects) and r = 3 (objects to arrange)
P(10,3) = 10! / (7)!
P(10,3) = 36,28,800 / 5,040 = 720
Example Calculation
Contest Ranking Problem
Problem: In a contest with 10 participants, how many ways can we arrange the top 3 winners (1st, 2nd, 3rd place)?
Given: n = 10 participants, r = 3 positions
Note: Order matters (1st ≠ 2nd ≠ 3rd), no repetition allowed
Solution
P(10,3) = 10! / (10-3)! = 10! / 7!
P(10,3) = 10 × 9 × 8
P(10,3) = 720 ways
There are 720 different ways to arrange the top 3 winners from 10 contestants.
Key Concepts
Without Repetition
Each element can only be used once
P(n,r) = n! / (n-r)!
Order Matters
ABC ≠ ACB ≠ BAC
Different arrangements = different permutations
Sequential Selection
First: n choices, Second: n-1 choices
Pattern: n × (n-1) × ... × (n-r+1)
Quick Examples
Mathematical Properties
P(n,0) = 1 (empty arrangement)
P(n,1) = n (single element)
P(n,n) = n! (all elements)
P(n,r) = C(n,r) × r!
Understanding Permutations without Repetition
What are Permutations without Repetition?
A permutation without repetition is an arrangement of distinct objects where the order matters and each object can only be used once. When you select r objects from n distinct objects and arrange them in a specific order, you're calculating P(n,r) or nPr.
Key Characteristics
- •Order matters: ABC ≠ BAC ≠ CAB
- •No repetition: Each element used only once
- •Selection and arrangement: Choose r from n and order them
- •Sequential choices: n choices first, n-1 second, etc.
Common Applications
- •Arranging people in a line or seating order
- •Contest rankings (1st, 2nd, 3rd place)
- •Password creation (without repeated characters)
- •License plate combinations
- •Committee officer positions
- •Scheduling distinct tasks in order
Memory Tip: Think "Position matters, No repeats!" - each position gets a unique item, and switching positions creates a different arrangement.
Mathematical Formula
Permutations of r objects from n distinct objects
Where:
- n: Total number of distinct objects
- r: Number of objects to arrange
- n!: n factorial (n × (n-1) × ... × 1)
- (n-r)!: (n-r) factorial
Alternative Formula:
- P(n,r) = n × (n-1) × ... × (n-r+1)
- Multiply r consecutive decreasing integers
- More efficient for large n, small r
- Avoids calculating large factorials
Example: Race Positions
Problem: 8 runners in a race. How many ways can 1st, 2nd, and 3rd place be awarded?
Solution: P(8,3) = 8!/(8-3)! = 8!/5! = 8×7×6 = 336 ways
Example: Password Creation
Problem: Create a 4-digit password using digits 1-9, no repetition allowed.
Solution: P(9,4) = 9!/(9-4)! = 9!/5! = 9×8×7×6 = 3,024 passwords