Primary formula
Without repetition C(n,r)=n!/[r!(n-r)!]; with repetition C(n+r-1,r).
Statistics calculator
Combination Calculator
Count groups selected from a larger set when rearranging the selected items does not create a new outcome.
Last reviewed: August 6, 2026Statistics calculation engine v2.0.0
Combination Calculator
Count groups selected from a larger set when rearranging the selected items does not create a new outcome.
Formula and assumptions
Input assumptions
Order is irrelevant. Items are distinct in no-repetition mode. n and r are nonnegative whole numbers.
Precision note
Statistics calculations use Decimal.js for deterministic arithmetic; displayed values are rounded only after the selected method is evaluated.
How NexaCalc evaluates the inputs
Parse
Finite numeric observations and method-specific inputs are validated locally.
Calculate
The selected Statistics method runs in a pure TypeScript engine.
Explain
The result includes substitutions, ordered data, supporting measures and method warnings.
Visualize
A tool-specific dataset visual appears only after a valid calculation.
What the Combination Calculator does
Use this Statistics page for committees, subsets and other selections where order does not matter.
Combination Calculator processes the entered values locally and exposes the selected convention, calculation steps and related values.
When to use the Combination Calculator
Use this Statistics page for committees, subsets and other selections where order does not matter.
Choose the mode that matches the data available and the question being answered before entering values.
Formula, variables and units
Without repetition C(n,r)=n!/[r!(n-r)!]; with repetition C(n+r-1,r).
- n is the number of available item types.
- r is the selection size.
- Each unordered group is counted once.
Statistical assumptions
The calculation is deterministic, but interpretation depends on whether the selected model and entered data fit the real situation.
- Order is irrelevant.
- Items are distinct in no-repetition mode.
- n and r are nonnegative whole numbers.
Step-by-step worked example
Use the default example inputs to reproduce the primary worked example. The result breakdown is produced by the same pure TypeScript engine used by the page.
- C(5,3)=10.
- C(10,2)=45.
- Choosing 3 from 5 types with repetition gives C(7,3)=35.
Additional examples and interpretation
The following examples show how data, event structure, count or method changes the result.
- C(5,3)=10.
- C(10,2)=45.
- Choosing 3 from 5 types with repetition gives C(7,3)=35.
Concept explanation
Count groups selected from a larger set when rearranging the selected items does not create a new outcome.
Use this Statistics page for committees, subsets and other selections where order does not matter.
How to read the statistics visual
The dynamic visual is a unordered selected group beside an available item pool. It appears only after a valid calculation and uses the current result.
Visual proportions are normalized for readability and are explanatory rather than a substitute for a full statistical plot.
Common mistakes
Check dataset parsing, method choice and denominator before using a result.
- Counting different orders as separate combinations.
- Using r greater than n without repetition.
- Using the repetition formula for distinct no-reuse selections.
Limitations and method-choice notes
Statistical conventions and model assumptions can change a result or its interpretation.
- Restrictions between choices require a more specific counting model.
- Large exact results are valid counts but may be difficult to interpret.
Educational and privacy note
This calculator provides educational statistics calculations and does not replace statistical consulting, instructor guidance or professional analysis.
Input values are processed in the browser calculator session and are not sent to an external statistics service.
Frequently asked questions
What does the Combination Calculator calculate?
It returns the exact number of unordered selections.
What inputs does the Combination Calculator require?
whole-number counts n and r
Which formula does the Combination Calculator use?
C(n,r)=n!/[r!(n-r)!] or C(n+r-1,r) with repetition
How should I interpret the Combination Calculator result?
It returns the exact number of unordered selections. Interpret it only under the assumptions stated on the page.
What assumptions are important for the Combination Calculator?
The standard combination formulas do not model dependencies or special restrictions among selected items.
What common mistake should I avoid in the Combination Calculator?
Select the mode that matches the question, keep units or probabilities consistent and do not treat a model output as stronger evidence than the inputs support.
How does the Combination Calculator differ from the Permutation Calculator?
The tools answer related but different questions. The Combination Calculator follows C(n,r)=n!/[r!(n-r)!] or C(n+r-1,r) with repetition, while the Permutation Calculator reports its own named quantity or model.
Can I use decimal values in the Combination Calculator?
Yes where the selected model permits them. Whole-number count fields reject decimals, while measurements, probabilities and dataset values accept finite decimals.
Does the Combination Calculator prove a statistical conclusion?
No. It performs a deterministic calculation; study design, data quality and subject-matter interpretation remain separate.
How does the Combination Calculator visual help?
The visual uses the current inputs and result to show the model structure, selected region or interval. It appears only after a valid calculation.
References
- OpenStax Introductory Statistics 2e, Measures of the Center of the Data. Relevance: mean, median and mode. Last verified: August 5, 2026. Source.
- OpenStax Introductory Statistics 2e, Measures of the Spread of the Data. Relevance: range, variance and standard deviation. Last verified: August 5, 2026. Source.
- NIST/SEMATECH e-Handbook, Measures of Scale. Relevance: statistical spread and scale measures. Last verified: August 5, 2026. Source.
- NIST/SEMATECH e-Handbook, Percentiles. Relevance: ordered data, ranks and percentile-method differences. Last verified: August 5, 2026. Source.
- OpenStax Introductory Statistics 2e, Two Basic Rules of Probability. Relevance: conditional, multiplication and addition rules. Last verified: August 6, 2026. Source.
- OpenStax Introductory Statistics 2e, Poisson Distribution. Relevance: discrete event-count probability models. Last verified: August 6, 2026. Source.
- OpenStax Introductory Statistics 2e, Confidence Intervals chapter review. Relevance: interval estimation, margin of error and sample size. Last verified: August 6, 2026. Source.
Statistics methods and references reviewed on August 6, 2026.
Educational disclaimer
This calculator provides mathematical results from the values, conventions and methods you enter. Verify important academic, engineering or professional work independently.