Permutation Calculator
Calculate the total number of ordered permutations (nPr) of choosing r elements from a set of n items.
Input Parameters
Live Result
Details
Steps
Identify permutation formula based on repetition mode
Expand descending sequence product
Calculate final permutation count
Quick Answer & Understanding Your Result
A Permutation Calculator computes ordered arrangements (nPr), calculating how many ways you can select and order r items from a set of n distinct elements where sequence matters.
Core Combinatorial Principles
- Order Matters (Arrangements): Arranging (A, B) is distinctly different from (B, A) (nPr = n! / (n - r)!).
- Without Replacement: Each selected item cannot be chosen again (r le n).
- With Replacement (n^r): Each selection is independent and items can be reused.
Practical Probability & Cryptography Uses
- PIN Codes & Passwords: Calculating keyspace size for 4-digit PINs or alphabetical passcodes.
- Race Results & Podiums: Determining possible 1st, 2nd, and 3rd place finishes in a competition.
Enter total items (n) and sample size (r) above to calculate permutations and step-by-step factorial breakdowns instantly.
Formula & Calculation Logic
Mathematical FormulaFactorial components:
- n (Total items): The full universe of distinct available elements.
- r (Selected items): Number of items arranged in sequential positions (0 le r le n).
- ! (Factorial): Product of all positive integers less than or equal to n.
Step-by-Step Guide
Enter Total Items (n)
Input the size of the original set (e.g. 10).
Enter Sample Size (r)
Input the number of positions to arrange (e.g. 3).
View Permutations & Steps
Receive total ordered arrangements and complete factorial expansion.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Podium Finish: 8 Runners, Top 3 Spots (8P3) | 8! ÷ (8 - 3)! = 8! ÷ 5! = 8 × 7 × 6 = 336. | ||
| Full Permutation: 5 items (5! = 120) | Provides instant quantitative evaluation for comparative practical analysis. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
NIST Digital Library of Mathematical Functions - Combinatorics
Khan Academy - Permutations and Combinations
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: nPr = n! ÷ (n - r)!
Last Reviewed: August 21, 2026
Primary Source: NIST Digital Library of Mathematical Functions - Combinatorics
Reference Document: Khan Academy - Permutations and Combinations
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.