Calculate ordered permutations and unordered combinations with safe integer checks.
How to use this calculator
Complete every field using the unit shown in its label. For permutations and combinations calculator, enter values that describe one consistent situation, then select Calculate. Review the formula and worked example before changing an assumption, and use Reset to restore the demonstrated starting values. For a useful second check, write down the input units and expected output unit before calculating. If the displayed answer differs from your estimate, verify the source values and selected options rather than adjusting the result manually; this keeps each scenario reproducible and makes later comparisons meaningful.
Formula & calculation method
P(n,r)=n!/(n-r)!; C(n,r)=n!/(r!(n-r)!)The calculator applies the stated relationship to your entries, preserves full precision during intermediate steps, and rounds only when displaying the results. Each output keeps its own unit so the answer can be checked against the question.
Worked example
For n=10 and r=3, permutations equal 720 and combinations equal 120. Their ratio is 3!, because each unordered selection has six orders.
Prepare your inputs and compare scenarios
Before entering numbers, define the question you want the calculator to answer and identify the source of each value. Review the fields below and use the requested unit or period for each one. When a value comes from a bill, document, or measurement, copy it with its meaning and units rather than converting from memory.
- Total items (n)
- Selected items (r)
Calculate your current or baseline situation first and keep the result. Then change only one input and recalculate to see its effect clearly. If a result looks surprising, check decimal separators, percentages, dates, and selected options before changing values. Comparing two realistic scenarios provides more useful context than relying on one isolated number.
Check the setup before checking the arithmetic
A correct calculation can still answer the wrong question. Identify what each number represents, choose the operation that matches the problem, and keep units attached to values. In multi-step work, preserve enough precision during intermediate steps and round only the final answer to a precision that suits the situation. Estimating a reasonable range beforehand makes typing and setup errors easier to spot.
Use the worked example to verify your understanding, then replace its values with your own. For geometry, confirm which dimensions are required and whether they describe diameter or radius, base or height. For algebra and statistics, inspect the complete input rather than focusing only on the result. If an answer is unexpected, simplify the problem to a small known case and rebuild it one step at a time.
- Write the known values, unknown value, and units before starting.
- Keep intermediate precision and round deliberately at the end.
- Test the method with a simple example whose answer you already know.
Understanding the result
Read the outputs together rather than selecting only the largest or smallest figure. The formula P(n,r)=n!/(n-r)!; C(n,r)=n!/(r!(n-r)!) defines exactly what is being measured; units and any status text are part of the answer. A result can be arithmetically correct while still being unsuitable if an input describes a different period, unit, or scope.
A practical comparison to try
Recreate the worked example first so you can confirm the units and operation. Then restore your real values and change only the most uncertain input. Comparing that baseline and revised case shows the sensitivity of the answer without mixing several effects.
Common mistakes to avoid
- Using combinations when order matters.
- Allowing r to exceed n.
- Treating a rounded floating-point value as an exact huge count.
Why this is useful
The tool makes the calculation repeatable and exposes the assumptions instead of hiding them inside a spreadsheet. Immediate recalculation supports like-for-like comparisons, while the formula, example, and limitations help you verify both the arithmetic and its interpretation.
Limitations to keep in mind
Inputs require whole numbers with 0≤r≤n. Results exceeding safe exact-integer range are rejected instead of being presented as exact factorial values.
Frequently asked questions
Why must the input units be consistent?
The formula (P(n,r)=n!/(n-r)!; C(n,r)=n!/(r!(n-r)!)) combines the entries directly. A different unit or period changes the meaning of the result unless it is converted first.
How can I check the result?
Repeat the worked example, then compare your own result with a simple manual estimate or the relevant source document.
Is this an official or professional determination?
No. It is an educational calculation from your inputs, not an official record, diagnosis, approval, or personalized recommendation.