Formula or algorithm and variables.
nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!)n is the total number of distinct items and r is the number selected without replacement.
Formula or algorithm and variables
nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!). n is the total number of distinct items and r is the number selected without replacement. Permutations count ordered selections: nPr = n!/(n−r)!; combinations ignore order: nCr = n!/(r!(n−r)!). Start by identifying the mathematical object represented by each input rather than treating every number as interchangeable. Keep the chosen mode visible when a tool has more than one interpretation. The calculation preserves exact integer arithmetic where the model is discrete and uses controlled numeric rounding where the result is real-valued. This makes the path from the input to the displayed answer inspectable and gives you a repeatable way to check a hand calculation.
Worked example
Selecting 2 people from 5 gives 20 ordered permutations but 10 unordered combinations. Reproduce the example by writing the formula, substituting the values, and retaining units or place-value labels through each step. For a list-based tool, sort or group the values before checking the result; for a base-conversion tool, expand each digit by its place value. When the result is a large integer, compare the exact string rather than a rounded approximation. A worked example is a verification aid, not a default recommendation, and it should be replaced with the values from the problem you are actually solving.
How to interpret the result
Choose nPr when position matters and nCr when the selected group is the outcome; both require r no greater than n. Read the primary result together with its supporting details and the selected mode. A mathematically correct result can still answer the wrong question if the inputs describe a different population, representation, unit, ordering rule, or sample space. Compare scenarios only when the definitions and assumptions remain the same. Use exact notation for factors, matrices, repeating decimals, and integer bases; use the displayed precision for real-number summaries and avoid implying more certainty than the inputs support.
Validation and edge cases
n and r must be whole numbers with 0 ≤ r ≤ n; values are limited to keep exact integer results displayable. Blank fields, malformed tokens, non-finite values, incompatible dimensions, invalid divisors, impossible probability ranges, and out-of-domain operations are rejected explicitly. Boundary values such as zero, one, a prime, a singular matrix, an empty remainder cycle, or a one-term sequence are tested according to the selected operation. If the output seems surprising, first verify the mode, the input order, the representation width, and whether a quantity was intended to be a count or a measurement.
Limitations and responsible use
The formulas assume distinct selectable items and no replacement. Repeated objects, replacement, restrictions, and conditional arrangements require a different model. This page is an explanatory calculator, not a substitute for a statistical design, numerical-analysis package, software type specification, laboratory model, or professional review. Results are processed locally in the browser and no entered values are sent to a calculation service. Keep the original inputs, mode, and assumptions with any copied result. For decisions involving health, finance, safety, experiments, or production systems, verify the model with current authoritative guidance and an independent method before acting.
Check assumptions and sources.
The formulas assume distinct selectable items and no replacement. Repeated objects, replacement, restrictions, and conditional arrangements require a different model.
Common questions.
What does the Permutation & Combination Calculator calculate?
Calculate nPr when order matters or nCr when it does not, using exact integer arithmetic.
Which formula or algorithm does it use?
nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!). n is the total number of distinct items and r is the number selected without replacement.
How should the result be interpreted?
Choose nPr when position matters and nCr when the selected group is the outcome; both require r no greater than n.
What limitations or edge cases matter?
n and r must be whole numbers with 0 ≤ r ≤ n; values are limited to keep exact integer results displayable. The formulas assume distinct selectable items and no replacement. Repeated objects, replacement, restrictions, and conditional arrangements require a different model.