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 |
If you like this calculator
Please help us simply by sharing it. It will help us a lot!
Related Calculators
Other calculators you might find useful.
Arithmetic Sequence Calculator
Calculate n-th term, sum, and generate sequence terms for arithmetic progressions
Molarity Calculator
Calculate solution concentration, moles, volume, and perform dilution calculations
Unit Rate Calculator
Calculate the rate per single unit from any ratio or comparison
Geometry Calculator
Calculate area, perimeter, volume, and surface area for 2D & 3D shapes
Ratio Calculator
Simplify ratios, compare ratios, scale values, and solve ratio problems with step-by-step solutions
Fibonacci Sequence Calculator
Calculate n-th Fibonacci number, sum of terms, and generate complete sequences
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