Interactive combinatorics studio

Factorial Calculation

Calculate factorials, double factorials, permutations and combinations with exact large-number results, scientific notation, worked steps and an interactive growth graph.

Exact BigInt results · worked steps · growth graph · counting tools

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Exact counting result

10! = 3,628,800

3628800

Scientific notation3.6288 × 10⁶
Number of digits7
Trailing zeros2
Stirling approximation error0.83%
Prime factorization2^8 × 3^4 × 5^2 × 7

Worked multiplication: 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Factorial growth graph

Switch between normal and logarithmic scales. The logarithmic view is clearly labeled and keeps rapid factorial growth readable.

nn!

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Why factorial numbers are remarkable

0! = 1

Why 0! equals 1

The empty arrangement has exactly one possible ordering, so 0! is defined as 1. This also keeps counting formulas consistent.

52!

A shuffled deck

A 52-card deck has 52! possible orders—far more than the estimated number of atoms on Earth.

100! → 158

Faster than exponential growth

Factorials grow so quickly that 100! already contains 158 digits.

What is a factorial?

For a non-negative integer n, n! is the product of every positive integer from n down to 1. By definition, 0! = 1.

n! = n × (n−1) × … × 2 × 1
0! = 1

Factorials and related counting formulas

Double factorial multiplies every second integer. Permutations count ordered selections, while combinations count selections where order does not matter.

nPr = n! ÷ (n−r)!
nCr = n! ÷ (r!(n−r)!)

Where factorials are used

Factorials appear in combinatorics, probability, binomial coefficients, Taylor series, algorithms, statistics and statistical physics.

Exact values and large inputs

Exact integers are calculated with BigInt up to n = 10,000. Very long results are shortened only on screen; copying and downloading preserve every digit.

n! ≈ √(2πn)(n/e)ⁿ
Γ(n+1) = n!

Frequently asked questions

What does n! mean?

It is the product n × (n−1) × … × 2 × 1 for a non-negative integer n.

Why is 0! equal to 1?

There is one way to arrange zero objects, and the value keeps permutation and combination formulas consistent.

Are the results exact?

Yes. The calculator uses BigInt, so supported non-negative integer results are exact rather than estimates.

What is a double factorial?

n!! multiplies every second positive integer: for example, 7!! = 7 × 5 × 3 × 1.

What is the difference between nPr and nCr?

nPr counts ordered selections; nCr counts selections where order does not matter.

Why use a logarithmic graph?

Factorials grow extremely quickly. A labeled log₁₀ scale makes values of very different sizes visible without pretending it is a normal scale.

What is Stirling’s approximation?

It estimates n! with √(2πn)(n/e)ⁿ and becomes increasingly accurate as n grows.

Can I copy a very large factorial?

Yes. Copy and Download include the complete exact integer even when the on-screen preview is shortened.