Factorial Calculator

Calculate exact factorials, permutations and combinations, and follow the multiplication behind every result.

Calculation inputs

Precision & interpretation

All three outputs use integer arithmetic. Very long results have a compact heading and an expandable exact value; copying and downloading retain the full result.

Calculation results

Factorial n!362880010! · exact integer

Permutations P(n,r)720

Combinations C(n,r)120

Digits in n!7

Step-by-step solution

Choose the formula
$n! = 1 × 2 × … × n; 0! = 1$
Substitute your inputs
$10! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10$
Calculate in order
$P = \frac{10!}{7!}; C = \frac{P}{3!}$
Write the result
$10! = 3628800; P = 720; C = 120$

Factor multiplication

Factor 11Factor 22Factor 33Factor 44Factor 55Factor 66Factor 77Factor 88Factor 99Factor 1010

Multiply every factor; the empty product is 1.

Exact arbitrary-size integer arithmetic. The compact heading does not round the full result.

Exact value & calculation details

3628800

Quantity	Value
n!	3628800
r!	6
(n−r)!	5040
P(n,r)	720
C(n,r)	120

Understand the calculation

Concept & meaning

A factorial counts ordered arrangements of distinct objects. Multiplying from 1 through n gives n!, with 0! defined as 1.

Formula & special cases

Permutations count ordered selections without replacement. Combinations count unordered selections, so each arrangement of the selected r objects is divided out.

Precision & limitations

All three outputs use integer arithmetic. Very long results have a compact heading and an expandable exact value; copying and downloading retain the full result.

n! = 1 \times 2 \times \dots \times n

Worked example from your inputs

10! · exact integer = 3628800

Frequently asked questions

Why is 0! equal to 1?

There is one way to arrange an empty selection. This value also keeps n! = n × (n−1)! consistent at n = 1.

Are large factorials rounded?

No. Up to n = 1,000 the calculation uses arbitrary-size integers. A shortened heading is only a presentation choice.

What if r is larger than n?

An ordered or unordered selection without replacement is impossible under this model. Enter 0 ≤ r ≤ n.

Does repetition count here?

No. P(n,r) and C(n,r) are the without-replacement counts. Use the Combinatorics Calculator for repeated selections.

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What does the Factorial calculate?

Calculate exact factorials, permutations and combinations, and follow the multiplication behind every result.

Which values should I enter?

Use the labeled fields above. Whole number from 0 to 1,000. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

n! = 1 × 2 × … × n; P(n,r) = n!/(n−r)!; C(n,r) = P(n,r)/r!

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$NIST official logo$ NIST DLMF · FactorialsFactorial notation and its gamma-function relationshipRead the reference $NIST official logo$ NIST DLMF · Combinatorial notationBinomial coefficients and falling factorialsRead the reference

References explain the mathematics; they do not imply institutional endorsement.