Probability
Combination Calculator
Calculate nCr, nPr, all subsets, and the combination count as a share of all subsets. The selection-board visual highlights chosen positions and contrasts one unordered group with its possible orderings.
Combinations nCr-
Permutations nPr-
All possible subsets-
Selected combinations as share of all subsets-
Decision view
Selection versus ordered arrangement
Selection versus ordered arrangementOne set of n distinct objects is shown as an unordered r-item selection and as its order-sensitive arrangements.
Exact scenario comparisonItems selected (r) changes while all other entered assumptions remain constant.
| Items selected (r) | Combinations nCr | Permutations nPr | All possible subsets | Selected combinations as share of all subsets |
|---|
How to use Combination Calculator
- Define whether objects are distinct.
- Decide whether order creates a different outcome.
- Enter n and r with zero less than or equal to r less than or equal to n.
Calculator guide
Understanding Combination Calculator
Combinations count selections when order does not matter; permutations count arrangements when it does. This distinction changes the answer even though both begin with the same n objects and r selections.
Calculation method
How the calculation works
Calculate unordered combinations, ordered permutations, and the full subset count using exact factorial relationships for practical input sizes. Use n! divided by r!(n-r)! for combinations, n! divided by (n-r)! for permutations, and two to the power n for all subsets.
Decision rule
Choose the counting model before calculating
Most errors come from classifying the outcome incorrectly.
Worked situations
Practical examples
- Choosing 5 members from 12 gives 792 unordered combinations.
- Assigning 5 ordered positions from 12 gives 95,040 permutations.
- The gap is 5! because each selected group has 120 internal orderings.
Better inputs
Useful tips
- Use combinations for committees and unordered hands.
- Use permutations for rankings, seat assignments, or ordered codes without replacement.
- For repeated choices or identical objects, use a different counting model.
Before relying on the result
Limitations and common mistakes
- The formulas assume distinct objects selected without replacement.
- Repeated selection, identical objects, circular arrangements, and constraints are excluded.
- Large exact integers can exceed floating-point precision in the runtime.
Reference
Key terms
- n
- Total number of available distinct objects.
- r
- Number of objects selected.
- Combination
- Selection in which internal order is ignored.
- Permutation
- Selection or arrangement in which order matters.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why is permutation larger than combination?
Each unordered selection can be arranged in r factorial orders.
Can r be zero?
Yes. There is exactly one empty selection.
Can r exceed n?
Not in selection without replacement.
Does this handle repeated digits?
No. Repetition requires a different formula.