Permutations & Combinations
Compute exact nPr permutations and nCr combinations with BigInt precision, multiplicative term cancellation, repetition options, and step-by-step formula explanations.
Set Parameters
Combinations C(52, 5)
Order does NOT matter (e.g. hand of cards)Permutations P(52, 5)
Order DOES matter (e.g. passwords, podium finishes)Mathematical Formula Breakdown
In-Browser Permutations and Combinations Computation
The Permutations and Combinations Calculator computes the number of ways to arrange (nPr) or select (nCr) a subset of r items from a total set of n distinct objects. Combinatorial counting principles are fundamental to probability theory, statistical sampling, cryptography, and algorithm optimization.
This utility distinguishes between ordered arrangements where position matters (permutations: nPr = n! / (n - r)!) and unordered selections where grouping matters (combinations: nCr = n! / (r! × (n - r)!)). It also calculates configurations with replacement and repetition.
Using arbitrary-precision BigInt arithmetic, the calculator evaluates combinatorial expressions for large numbers without floating point overflow. Complete step-by-step factorial expansion formulas are displayed for every calculation.
Core Tool Capabilities
- Simultaneous nPr and nCr Evaluation: Computes permutations (order matters) and combinations (order does not matter) simultaneously for instant comparison.
- Repetition and Replacement Options: Supports combinations with repetition (n + r - 1)! / (r! × (n - 1)!) and permutations with replacement (n^r).
- Step-by-Step Formula Expansions: Displays the complete algebraic cancellation steps showing how numerator and denominator factorials simplify.
- Arbitrary-Precision BigInt Math: Calculates exact values for large combinatorial problems without rounding to scientific notation.
Local Browser Privacy
Your files are processed locally on your device. Files are not uploaded, transmitted, or stored on remote servers.
Once the page has loaded, you can disconnect your network connection and continue using this tool offline.
Target Use Cases
- • Probability and Gaming Theory
- • Cryptography and Password Security
- • Team and Committee Selection
- • Academic Coursework
Technical Specifications and Platform Limits
Standard mathematical permutation and combination definitions
Validates non-negative integer input parameters
Calculates results locally with zero external network requests
Your mathematical problems and parameters are never stored
Step-by-Step Workflow
Enter Total Items (n)
Input the total number of items in the set as a positive integer.
Enter Items to Choose (r)
Input the number of items being selected or arranged (where r ≤ n).
Choose Order and Repetition Rules
Specify whether order matters (Permutation vs Combination) and if repetition is allowed.
Inspect Results and Step Formulas
Review the calculated total possible arrangements and step-by-step factorial cancellations.
Practical Real-World Use Cases
Probability and Gaming Theory
Calculating lottery odds, poker hand combinations, and dice rolling probabilities.
Cryptography and Password Security
Estimating total keyspace entropy based on character set permutations of length r.
Team and Committee Selection
Determining the number of ways to pick a 5-person committee from a group of 20 candidates.
Academic Coursework
Students and educators verifying homework solutions in discrete mathematics and statistics.
Frequently Asked Questions
- What is the main difference between permutations and combinations?
- In permutations, order matters (e.g., race podium finish). In combinations, order does not matter (e.g., fruit salad ingredients).
- Why does nCr always yield a whole number?
- Combinations count distinct physical groupings of objects, which is always an integer according to combinatorial identity.
- What is the value of nC0 and nCn?
- Both nC0 (choosing 0 items) and nCn (choosing all items) always equal 1.
- Are my calculations sent to a remote server?
- No. All factorial and combinatorial calculations execute entirely inside your local browser memory.
- Can the tool compute combinations where items can be repeated?
- Yes. Enabling repetition calculates combinations with replacement using the formula (n + r - 1)! / (r! × (n - 1)!).
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