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What is a Permutation Calculator?
This calculator computes permutations (nPr) and arrangements where order matters, plus combinations and factorials for comparison.
This calculator computes permutations (nPr) and arrangements where order matters, plus combinations and factorials for comparison.
Key Concepts
- Permutations (nPr): n! ÷ (n-r)! - Order matters (arrangements)
- Combinations (nCr): n! ÷ (r! × (n-r)!) - Order doesn't matter (selections)
- Factorial (n!): n × (n-1) × (n-2) × ... × 1
Example: Arranging 3 people from 5 in specific positions: 5P3 = 60 ways. Just selecting 3 people: 5C3 = 10 ways.
Calculation Results
Enter n and r values, then select a calculation type to see results.
Calculation Details
Permutation vs Combination
Common Arrangements
Related Math Tools
How to Use the Permutation Calculator
- Enter the total number of items (n) and the number of items to arrange (r).
- Click "nPr" for permutations (order matters) or "nCr" for combinations (order doesn't matter).
- Use "n!" or "r!" buttons to calculate factorials of your input values.
- Try the preset examples to understand common permutation scenarios.
This calculator shows both permutation and combination results to help you understand the difference.
When Order Matters (Permutations)
Use Permutations (nPr) when:
- Arranging people in line: First, second, third positions
- Race finishing order: Gold, silver, bronze medals
- Password creation: ABC123 ≠ 321CBA
- Job assignments: President, VP, Secretary roles
- Sequence matters: Steps in a process
Formula: nPr = n! ÷ (n-r)!
Real-World Permutation Examples
- Sports tournaments: 8 teams, top 3 positions (8P3 = 336)
- Door codes: 4 digits from 10 numbers (10P4 = 5,040)
- Playlist order: 20 songs, play first 5 (20P5 = 1,860,480)
- Seating arrangements: 6 people, 6 chairs (6P6 = 720)
- License plates: 3 letters from 26 (26P3 = 15,600)
- Work schedules: 10 employees, 4 shifts (10P4 = 5,040)
Frequently Asked Questions
Ask yourself: "Does the order matter?" If yes, use permutations (nPr). If you can rearrange the selected items and get the same result, use combinations (nCr). For example, choosing a team captain, vice-captain, and secretary from 10 people uses permutations because each role is different. Choosing any 3 people for a committee uses combinations because the roles are the same.
Permutations count all possible arrangements of the selected items, while combinations only count the selection itself. For example, selecting A, B, C can be arranged as ABC, ACB, BAC, BCA, CAB, CBA (6 permutations) but represents only 1 combination. Permutations = Combinations × r! because there are r! ways to arrange r items.
When r = n, you're arranging all available items, so nPr = n!. For example, arranging all 5 people in 5 chairs gives 5P5 = 5! = 120 different arrangements. This is the total number of ways to order all items in the set.
This calculator handles permutations without repetition (each item used once). For permutations with repetition allowed, you'd use n^r. For example, a 4-digit PIN using digits 0-9 would be 10^4 = 10,000 possibilities. Permutations with some repeated items use the formula n!/(n1! × n2! × ...) for items with n1, n2... repetitions.