Statistics

Permutation and Combination Calculator

Determine the total number of permutations (ordered) and combinations (unordered) for selections.

Reviewed by the Basic Math Tools Editorial Team

Last updated: June 2026

Formula

nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}

Input Parameters

Math Keypad

Result

Calculated Answer

Provide inputs to solve.

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BasicMathTools STEM Editorial Team Verified

Reviewed by Applied Math & Finance Specialists • Editorial Policy

Updated: July 2026 • Fact-Checked

What Is the Permutation and Combination Calculator?

The Permutation and Combination Calculator provides a reliable, step-by-step mathematical framework for calculating permutation and combination problems in Statistics. Whether you are a student working through homework, an educator creating lesson materials, or a working professional, this calculator eliminates manual arithmetic errors while displaying the exact logic used at every stage.

At the core of this tool is the formula nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}. Understanding how each parameter interacts within this equation is key to mastering permutation and combination calculations.

How to Use the Permutation and Combination Calculator

Using this calculator is straightforward. Enter your known values into the fields below, and the solver will compute the result immediately. Here is what each input represents:

Total Items (n) Example input: e.g. 5

Selected Items (r) Example input: e.g. 3

Formula Used

nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}

Worked Example

Worked Example: Step-by-Step Permutation and Combination Problem

Problem: Calculate the result for Permutation and Combination given the inputs: Total Items (n) = 5, Selected Items (r) = 3.

1. Collect and Verify Inputs

Identify the values provided for the problem (Total Items (n) = 5, Selected Items (r) = 3). Ensure all measurement units are consistent before calculating.

2. Substitute Into the Primary Formula

Substitute the values into the mathematical equation: nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}.

nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}

3. Perform Intermediate Arithmetic Operations

Evaluate exponential, multiplication, division, or algebraic operations following standard order of operations (PEMDAS/BODMAS).

4. Format and Round Final Output

Round the final calculated numerical output to standard decimal precision (e.g. 2-4 decimal places).

Final Result Result verified and calculated via Permutation and Combination Calculator

How to Calculate Permutation and Combination Step-by-Step

Understanding the underlying solution workflow helps build mathematical intuition:

Identify and note down the given values for: Total Items (n), Selected Items (r).

Set up the primary formula: \(nPr = \frac{n!}{(n-r)!}, \ nCr = \frac{n!}{r!(n-r)!}\). Substitute the identified values into their respective positions.

Complete the statistical calculations (e.g., sum, mean, or computing variance and probability) from the dataset.

Round the final calculated answer to the required decimal accuracy or significant figures.

Real-World Applications of Permutation and Combination Calculator

Scientific Research & Hypothesis Testing

Researchers use statistical measures to analyze experimental sample groups, test significance, and publish peer-reviewed findings.

Business Intelligence & Market Research

Data analysts measure customer retention, average order value, variance, and trend forecasts to guide corporate strategy.

Medical Clinical Trials

Used by epidemiologists to evaluate treatment efficacy, drug trial safety margins, and disease spread modeling.

Common Pitfalls & Mistakes to Avoid

Inputting Incorrect Units or Unmatched Intervals

Always verify that all input values use consistent units (e.g. converting percentages to decimals or matching annual vs. monthly rate periods) before computing.

Neglecting Order of Operations (PEMDAS/BODMAS)

When performing manual checks, calculate operations inside parentheses first, followed by exponents, multiplication/division, and finally addition/subtraction.

Rounding Intermediate Values Too Early

Keep full floating-point precision throughout intermediate calculation steps. Only round the final answer to your desired decimal accuracy.

Key Terminology Glossary

Total Items (n) The n parameter specified in the Permutation and Combination Calculator. Enter numeric values to compute results.

Selected Items (r) The r parameter specified in the Permutation and Combination Calculator. Enter numeric values to compute results.

Expert Tips for Permutation and Combination Calculator

Use Permutations (nPr) when the order of items matters (e.g., combinations of a safe), and Combinations (nCr) when order does not matter (e.g., lottery numbers).
Because order matters, the number of permutations of a subset is always greater than or equal to the number of combinations.
For combinations, the formula is n! / (r!(n-r)!), which divides out the redundant permutations of the selected items.

About the Permutation and Combination Calculator

The Permutation and Combination Calculator is maintained by Basic Math Tools, an educational platform committed to providing accurate STEM and financial computing tools. Every tool processes calculations transparently in your browser for privacy and speed.

If you have suggestions or request additional features, please review our Editorial Policy or contact our math team.

Frequently Asked Questions

What is the difference between permutation and combination?

In permutations, the order of selection matters (e.g. combination lock). In combinations, order does not matter (e.g. picking a hand of cards).

Does the Permutation and Combination Calculator use population or sample formulas?

The calculator supports both. When the distinction matters (e.g., standard deviation), the interface will either let you choose or clearly indicate which formula is being applied. Sample formulas use N-1 (Bessel's correction); population formulas use N.

How do I enter my data into the statistics calculator?

Enter your data values separated by commas (e.g., 12, 15, 18, 22, 25). The calculator will parse the values automatically. You can also paste data directly from a spreadsheet.

What statistical measures does this tool calculate?

Determine the total number of permutations (ordered) and combinations (unordered) for selections. The results include the computed values along with the formulas used, so you can verify the methodology.

Can I use this for academic research?

The mathematical computations are accurate and suitable for coursework and preliminary analysis. For published research, we recommend verifying results with statistical software like R or SPSS, and always report your methodology clearly.