Factorial Calculator
Compute n factorial (n!) for whole numbers from 0 to 170, with examples, the 0! = 1 rule and why large factorials overflow.
Results
10!
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Educational estimates · not financial advice · free, no signup
Formula / how it works
n! = n × (n − 1) × … × 2 × 1, with 0! = 1. For example 5! = 120 and 10! = 3,628,800. Results above 170! exceed the range of standard floating-point numbers.
Detailed guide
What it is, how to use it, how to read the numbers, common mistakes, a worked example, formulas, and FAQs.
What this calculator is
The factorial of a whole number n is the product of every positive integer up to n. It counts the number of ways to arrange n distinct items in a row: three books can be ordered 3! = 6 ways. Factorials appear throughout permutations, combinations, probability, series expansions and algorithm analysis. They grow extremely fast. 10! is already 3.6 million, 20! is about 2.4 × 10¹⁸, and 170! is about 7.3 × 10³⁰⁶. Beyond 170, the value exceeds the maximum a standard double-precision number can hold, so this calculator limits n to 170.
When to use / who it helps
- You need n! for a permutations or probability problem.
- You are checking homework on series, binomial coefficients or counting.
- You want to see how fast factorials grow.
- You are testing a program that computes factorials.
- You need a value for the denominator of a Taylor series term.
How to use
Steps
- Enter a whole number n from 0 to 170.
- Read n! in the result.
- For counting problems, decide whether order matters before using a factorial.
- Use combinations instead if you are choosing a subset.
- For large values, read the result in scientific notation.
- Use exact big-number software for n above 170.
How to interpret results
- n! counts the orderings of n distinct objects.
- Large results may be rounded floating-point values rather than exact integers.
- 0! equals 1 by convention, matching the empty product.
- Dividing factorials gives permutations and combinations.
- Factorials of negative numbers are not defined in basic arithmetic.
Common mistakes & gotchas
- Entering a decimal or negative number.
- Assuming 0! is 0.
- Expecting exact digits for very large n.
- Using factorials where order does not matter.
- Confusing n! with n squared.
Worked example
Worked example — 5! and 10!
- 5! = 5 × 4 × 3 × 2 × 1 = 120.
- 10! = 10 × 9 × … × 1 = 3,628,800.
- Check: 10! = 10 × 9! and 9! = 362,880.
- As a counting example, 5 people can be arranged in a line in 120 different orders.
Result: 5! = 120 and 10! = 3,628,800.
To load the form's own demo values, tap Try example above the Calculate button.
How to calculate / formula
n! = n × (n − 1) × … × 2 × 1, with 0! = 1. For example 5! = 120 and 10! = 3,628,800. Results above 170! exceed the range of standard floating-point numbers.
Frequently asked questions
What is 0 factorial?
1, by convention, since there is exactly one way to arrange no objects.
Why is the limit 170?
171! is larger than the largest double-precision number, so the result would overflow.
What is a factorial used for?
Counting arrangements, permutations, combinations and in series like e^x.
How is a factorial different from a power?
A factorial multiplies decreasing integers; a power multiplies the same number repeatedly.
Can I find the factorial of a decimal?
Not with this tool. The gamma function extends factorials to non-integers.
How do I get combinations from factorials?
nCr = n! ÷ (r!(n − r)!).