Permutations Calculator
Result:
Calculate permutations P(n,r) with our free online calculator. Perfect for arrangements, sequences, and order-dependent selection problems in combinatorics and probability.
What are Permutations?
Permutations count the number of ways to arrange r items from a set of n items where order matters. Different arrangements of the same items are counted as different permutations.
Permutation Formula
P(n,r) = n! / (n-r)!P(n,n) = n! (all items)P(n,1) = n (single item)
Permutation Properties
Key Properties
- Order matters: ABC ≠ BCA
- P(n,0) = 1 (empty arrangement)
- P(n,1) = n (choose one)
- P(n,n) = n! (all items)
- P(n,r) ≥ C(n,r) (more than combinations)
Relationship to Combinations
- P(n,r) = C(n,r) × r!
- Permutations: Order-dependent
- Combinations: Order-independent
- Factor difference: r! times more permutations
- Use case: Arrangements vs. selections
Step-by-Step Examples
Problem: How many ways to arrange 3 people from a group of 5?
Solution: P(5,3) = 5! / (5-3)! = 5! / 2!
Calculation:
P(5,3) = (5 × 4 × 3 × 2 × 1) / (2 × 1)
P(5,3) = 120 / 2 = 60
Alternative: P(5,3) = 5 × 4 × 3 = 60
Answer: 60 different arrangements
Problem: Arrange 2 letters from {A, B, C, D}
Formula: P(4,2) = 4! / (4-2)! = 4!/2! = 24/2 = 12
All arrangements:
- AB, AC, AD
- BA, BC, BD
- CA, CB, CD
- DA, DB, DC
Count: 4 × 3 = 12
Pattern: First position has 4 choices, second has 3
Common Permutation Values
| P(n,r) | Formula | Value | Description |
|---|---|---|---|
| P(3,1) | 3!/2! | 3 | Choose 1 from 3 |
| P(3,2) | 3!/1! | 6 | Arrange 2 from 3 |
| P(3,3) | 3!/0! | 6 | All arrangements |
| P(4,2) | 4!/2! | 12 | Arrange 2 from 4 |
| P(5,2) | 5!/3! | 20 | Arrange 2 from 5 |
| P(5,3) | 5!/2! | 60 | Arrange 3 from 5 |
| P(6,3) | 6!/3! | 120 | Arrange 3 from 6 |
| P(10,2) | 10!/8! | 90 | Arrange 2 from 10 |
Real-World Applications
Sports & Competition
- Race finishing positions
- Tournament rankings
- Team batting orders
- Medal ceremonies (1st, 2nd, 3rd)
- Draft pick orders
Organization & Planning
- Seating arrangements
- Meeting presentation order
- Work shift scheduling
- Committee officer positions
- Event program ordering
Security & Technology
- Password combinations
- Access code sequences
- Cryptographic keys
- Algorithm permutations
- Network routing paths
Frequently Asked Questions (FAQ)
The key difference is whether order matters:
Permutations (Order Matters)
- Formula: P(n,r) = n!/(n-r)!
- Example: Race winners: 1st, 2nd, 3rd
- ABC ≠ BCA: Different arrangements
- Use when: Position/order is important
Combinations (Order Doesn't Matter)
- Formula: C(n,r) = n!/(r!(n-r)!)
- Example: Selecting team members
- ABC = BCA: Same selection
- Use when: Just selecting items
Use permutations when order or position matters:
- Sequences: When items must be in a specific order
- Rankings: 1st place is different from 2nd place
- Arrangements: Seating plans, lineups, schedules
- Positions: President, VP, Secretary roles
- Passwords/Codes: Where sequence matters
- Key phrase: If swapping items gives a different result, use permutations
When r = n, you're arranging all available items:
- Formula becomes: P(n,n) = n!/(n-n)! = n!/0! = n!
- Result: All possible arrangements of n items
- Example: P(4,4) = 4! = 24 ways to arrange 4 people
- Real-world: All possible orders for a complete lineup
- Growth: Grows very quickly (4! = 24, 5! = 120, 10! = 3,628,800)
Yes! There are different types of permutations:
- Without repetition (this calculator): P(n,r) = n!/(n-r)!
- With repetition: Each position can use any of the n items
- Formula with repetition: nʳ (n choices for each of r positions)
- Example: 4-digit PIN using digits 0-9: 10⁴ = 10,000 possibilities
- Use case: Passwords, license plates, phone numbers
Our calculator handles up to n=20, but values grow very quickly:
- P(10,5) = 30,240 (manageable size)
- P(15,10) = 10,897,286,400 (over 10 billion)
- P(20,20) = 20! ≈ 2.4 × 10¹⁸ (astronomical)
- Growth rate: Permutations grow faster than combinations
- For larger values: Use logarithms or specialized software
- Practical limit: Most real-world problems use smaller values
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