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Enter your values, then click Calculate result.How this calculator helps
Count selections or ordered arrangements of distinct items without replacement using exact integer arithmetic for 0 ≤ r ≤ n ≤ 1000. The available count n describes distinguishable items, and the selected count r describes how many are used. The model selects each item at most once. Repeated picks with replacement, indistinguishable objects and strings with repeated letters require other counting formulas. Define those conditions before selecting a mode rather than using the largest returned number as a universal count.
How to use it
- 1
Choose the correct input basis for permutation and combination calculator and enter the values described below. The demonstration defaults are examples, not independently verified personal measurements.
- 2
Review does order matter? before submitting. Match the selected units and roles to the original source record, including any signs or percentage conventions.
- 3
Click Calculate result to submit the current inputs. Editing a field preserves the previous submitted output until you calculate again; the status message identifies that pending change.
- 4
Compare the labeled result with the worked example and independent verification checks. Review counts and probabilities before copying it into another worksheet.
Formula and methodology
nPr = n!/(n−r)!. nCr = n!/[r!(n−r)!]. Permutations distinguish order; combinations count each unordered selection once.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
Choosing 3 people from 10 for one committee gives 10C3 = 120 combinations. Assigning three different roles to those same people gives 10P3 = 720 permutations. The factor between them is 3! = 6, representing the possible role orders for each selected trio.
How to interpret your result
The calculation multiplies integer factors using BigInt. For combinations it uses the smaller of r and n−r and divides at each stage where the recurrence is exact. This avoids floating-point factorial overflow and preserves large integer digits. The output is exact for the entered counts, not a rounded scientific approximation. The upper bound of one thousand keeps browser work and output size controlled.
For different inputs or formulas, use Probability Calculator; Hypergeometric Calculator.
Related questions this calculator covers
- permutation calculator
- permutations calculator
- ncr calculator
- combination calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Unordered | 10C3 = 120. |
| Ordered | 10P3 = 720. |
| Symmetry | 10C3 equals 10C7. |
Common mistakes to avoid
- Ignoring whether positions make order matter.
- Using a without-replacement formula for repeated picks.
- Presenting a count as a probability without a total outcome space.
For small examples, list outcomes directly. Check nCr = nC(n−r), and verify nPr = nCr × r!. Boundary tests with r = 0 and r = n provide useful checks without enumerating large sets. If constraints prohibit some arrangements, those outcomes need a separate model rather than being silently included.
Authoritative reference. Method references reviewed on 5 October 2026. SolvePilot supplies the original examples and bounded browser implementation. Review by Mohammad Qasim covers editorial scope and arithmetic, not individual professional approval. The cited reference provides method or unit context rather than certifying the entered measurements or assumptions.What can affect the result?
Distinct items and replacement
The available count n describes distinguishable items, and the selected count r describes how many are used. The model selects each item at most once. Repeated picks with replacement, indistinguishable objects and strings with repeated letters require other counting formulas. Define those conditions before selecting a mode rather than using the largest returned number as a universal count.
Does order matter?
Use combinations when a selected group is the same regardless of listing order. Use permutations when positions, ranks or roles make arrangements different. Selecting three committee members is unordered; assigning president, secretary and treasurer is ordered. The numeric inputs can be identical while the intended count differs by r factorial. The interface labels the chosen interpretation in the result.
Exact integer arithmetic
The calculation multiplies integer factors using BigInt. For combinations it uses the smaller of r and n−r and divides at each stage where the recurrence is exact. This avoids floating-point factorial overflow and preserves large integer digits. The output is exact for the entered counts, not a rounded scientific approximation. The upper bound of one thousand keeps browser work and output size controlled.
Boundary cases
Selecting no items gives one empty selection or arrangement, so r = 0 returns one. Choosing all n items gives one combination, while permuting all n items gives n factorial. Available and selected counts must be whole numbers satisfying zero through one thousand and r no greater than n. A request to choose more distinct items than exist is rejected rather than treated as a probability of zero.
Counts and probabilities
A count is not itself a probability. To form a probability, identify equally likely outcomes and divide a favorable count by the total under the same model. Unequal weights, dependent events and constrained assignments can break a simple counting approach. This page does not generate every arrangement, impose custom restrictions or supply repeated-trial distribution assumptions. Use the separate probability tools only after defining the experiment.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
A count is not itself a probability. To form a probability, identify equally likely outcomes and divide a favorable count by the total under the same model. Unequal weights, dependent events and constrained assignments can break a simple counting approach. This page does not generate every arrangement, impose custom restrictions or supply repeated-trial distribution assumptions. Use the separate probability tools only after defining the experiment.
Sources and review information
Frequently asked questions
Which mode is for a committee?＋
Combinations, when member order and roles do not matter.
Which mode is for ranked positions?＋
Permutations, when the same selected items in different positions count separately.
Can I use replacement?＋
No. Each distinct item is selected at most once in this worksheet.
Why does choosing zero give one?＋
There is one empty selection, the standard combinatorial boundary case.
Are very large answers rounded?＋
No. The supported counts use exact BigInt arithmetic and display the full integer.