Permutation Calculator
Calculate nPr permutations with step-by-step solutions
nPr = n! / (n - r)!
The number of ways to arrange r items selected from n distinct items (order matters)
Total number of distinct items
Must be less than or equal to n
Quick Examples
Invalid Input
P (Permutations)
ways to arrange items from distinct items
Permutation (Order Matters)
P
Combination (Order Doesn't Matter)
C
Key Insight: Permutations are always greater than or equal to combinations because permutations count each unique ordering as a different arrangement. P = ! × C
Permutation Table: 1 to
| n | r | nPr |
|---|---|---|
Step-by-Step Breakdown
Permutation Calculation
Formula Applied
P = ! / ( - )! = /
Common Permutation Values
| n | r | nPr | Example |
|---|---|---|---|
| 3 | 2 | 6 | 2-letter arrangements from ABC |
| 4 | 3 | 24 | Podium positions in a race |
| 5 | 3 | 60 | 3-digit codes from 5 numbers |
| 10 | 3 | 720 | Top 3 from 10 contestants |
| 26 | 3 | 15,600 | 3-letter codes from alphabet |
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About Permutation Calculator
What is a Permutation?
A permutation is an arrangement of objects in a specific order. Unlike combinations, the order of selection matters in permutations. The notation nPr (also written as P(n,r)) represents the number of ways to arrange r items selected from n distinct items.
Permutation Formula
nPr = n! / (n - r)!
Where:
- n = total number of distinct objects
- r = number of objects to arrange (r ≤ n)
- ! = factorial (product of all positive integers up to that number)
How to Calculate Permutations
Example: 5P3
5P3 = 5! / (5-3)! = 5! / 2! = (5 × 4 × 3 × 2 × 1) / (2 × 1) = 120 / 2 = 60
Shortcut Method
Multiply n down to (n-r+1): 5P3 = 5 × 4 × 3 = 60
Common Permutation Values
| n | r | nPr | Example |
|---|---|---|---|
| 3 | 2 | 6 | 2-letter arrangements from ABC |
| 4 | 3 | 24 | Podium positions in a 4-person race |
| 5 | 3 | 60 | 3-digit codes from 5 numbers |
| 6 | 2 | 30 | Assigning 2 tasks to 6 people |
| 10 | 3 | 720 | Top 3 winners from 10 contestants |
Permutations vs Combinations
| Aspect | Permutation (nPr) | Combination (nCr) |
|---|---|---|
| Order | Matters | Doesn't matter |
| Formula | n!/(n-r)! | n!/[r!(n-r)!] |
| Example | Arranging books on a shelf | Choosing team members |
| Value | Always ≥ nCr | Always ≤ nPr |
Real-World Applications
- Lock Combinations - PIN codes, passwords (order matters)
- Race Results - First, second, third place rankings
- Scheduling - Assigning time slots to events
- Seating Arrangements - Arranging people in specific seats
- License Plates - Creating unique alphanumeric sequences
Special Cases
- nP0 = 1 (There's exactly one way to arrange 0 items)
- nPn = n! (Arranging all n items)
- nP1 = n (Choosing one item from n items)
Tips for Problem Solving
- Identify if order matters (permutation) or not (combination)
- Determine n (total items) and r (items to arrange)
- Apply the formula nPr = n! / (n-r)!
- Use the shortcut method for faster calculation