Free Permutation & Combination Calculator Online
nPr and nCr calculations with factorial steps and formula explanation
The Permutation & Combination Calculator computes P(n,r) and C(n,r) simultaneously, showing the factorial expansion at each step. Permutations count ordered arrangements; combinations count unordered selections. Probability students, exam takers, and game designers use this tool to count possibilities without manually computing large factorials.
Frequently Asked Questions
About the Permutation & Combination
Counting principles are the foundation of probability theory. Permutations (P(n,r) = n!/(n−r)!) count the number of ways to arrange r items from a set of n where order matters. Combinations (C(n,r) = n!/(r!×(n−r)!)) count the number of ways to choose r items where order does not matter. The distinction is critical: choosing a 3-member team from 8 people gives C(8,3) = 56 ways, but assigning them first, second, and third place gives P(8,3) = 336 ways.
This calculator shows the full factorial expansion, making it clear how the denominator cancels terms. For large n and small r, computing P or C directly via factorials can overflow — the calculator handles this gracefully.
Applications include probability problems (lottery odds, card game hands), combinatorics in algorithm analysis (Big-O notation), DNA sequence counting in bioinformatics, and scheduling problems in operations research.
Formula Used
P(n,r) = n! / (n-r)!
C(n,r) = n! / (r! × (n-r)!)
When Should You Use This?
The Permutation & Combination is ideally suited for students, teachers, engineers, and scientists who need to perform quick, accurate calculations related to general calculations. Use this tool when you need to verify figures, compare different scenarios, or get a precise answer without manual computation errors.
What Does The Result Mean?
The calculated output provides an instant, accurate resolution to your input parameters. You can use these results directly for your planning, assignments, or professional tasks, knowing they are based on standardized formulas.
Example Calculation
How many ways can you choose 3 items from 8 (ordered vs. unordered)?
📥 Inputs
- n = 8 (total items)
- r = 3 (items chosen)
🔢 Calculation Steps
- 1Permutation P(8,3) — order matters:
- 2P(8,3) = 8! / (8−3)! = 8! / 5!
- 3= (8 × 7 × 6 × 5!) / 5! = 8 × 7 × 6 = 336
- 4Combination C(8,3) — order does not matter:
- 5C(8,3) = 8! / (3! × 5!) = 336 / 3! = 336 / 6 = 56
Limitations of this Calculator
- Results are based purely on the mathematical relationship of the inputs provided.
- Does not account for edge cases or extreme outlier values that fall outside standard formula constraints.
- Calculated outputs should be double-checked against your specific real-world requirements before finalizing important decisions.
How to Use the Permutation & Combination
- 1Enter your values into the Permutation & Combination input fields above.
- 2Review the input labels to ensure you are using the correct units.
- 3Click the "Calculate" button to get your instant result.
- 4Use the step-by-step breakdown to understand how the result was calculated.
- 5Export or copy your result to use in reports or share with others.
Tips & Best Practices
- Double-check your input units before calculating — using the wrong unit is the most common source of errors.
- Bookmark this Permutation & Combination for quick access next time you need it.
- Use the share button to send your results to a colleague or save them for later reference.
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