Factorial Calculator

About Factorial Calculator

Calculate n! for any non-negative integer with exact BigInt results up to 1000! (numbers with thousands of digits). Also supports double factorial (n!!), subfactorial (!n), and Stirling's approximation. The full multiplication expansion is shown for smaller values.

What Is a Factorial?

The factorial of n (written n!) is the product of all positive integers from 1 to n:

n! = n × (n-1) × (n-2) × ... × 2 × 1

By convention, 0! = 1. This makes formulas for permutations, combinations, and the binomial theorem work consistently.

Factorial Values Reference

n	n!	Digits
0	1	1
1	1	1
2	2	1
3	6	1
4	24	2
5	120	3
6	720	3
7	5,040	4
8	40,320	5
9	362,880	6
10	3,628,800	7
12	479,001,600	9
15	1,307,674,368,000	13
20	2,432,902,008,176,640,000	19
25	~1.551 × 10^25	26
50	~3.041 × 10^64	65
100	~9.333 × 10^157	158
1000	~4.024 × 10^2567	2,568

Factorials grow faster than exponential functions. 100! has 158 digits, while 2^100 has only 31 digits.

Why Does 0! Equal 1?

There are several ways to see why 0! must be 1:

• Combinatorial argument: There is exactly one way to arrange zero items - do nothing. So 0! = 1.
• Recursive definition: n! = n × (n-1)!. For this to work at n=1: 1! = 1 × 0!, so 0! = 1.
• Binomial coefficient: nC0 = n!/(0! × n!) = 1. This requires 0! = 1.
• Empty product: A product of no factors is defined as the multiplicative identity, which is 1.

Double Factorial (n!!)

The double factorial multiplies every other integer down from n:

• Odd n: n!! = n × (n-2) × (n-4) × ... × 3 × 1
• Even n: n!! = n × (n-2) × (n-4) × ... × 4 × 2

n	n!! (odd)	n	n!! (even)
1	1	2	2
3	3	4	8
5	15	6	48
7	105	8	384
9	945	10	3,840
11	10,395	12	46,080

Double factorials appear in physics (quantum mechanics, spherical harmonics) and in the formulas for the volume and surface area of n-dimensional spheres.

Subfactorial (!n) - Derangements

The subfactorial !n counts derangements - permutations where no element is in its original position.

Example: For {1, 2, 3}, the derangements are {2, 3, 1} and {3, 1, 2}. So !3 = 2.

!n = n! × (1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)

n	n!	!n	!n/n!
1	1	0	0%
2	2	1	50%
3	6	2	33.3%
4	24	9	37.5%
5	120	44	36.7%
10	3,628,800	1,334,961	36.8%

The ratio !n/n! converges quickly to 1/e ≈ 0.3679. This means about 36.8% of all random permutations are derangements, regardless of the size of the set.

Stirling's Approximation

For large n, computing n! exactly is impractical. Stirling's formula provides a close approximation:

n! ≈ √(2πn) × (n/e)^n

n	Exact n!	Stirling's	Error
5	120	118.02	1.65%
10	3,628,800	3,598,695	0.83%
20	2.432 × 10^18	2.423 × 10^18	0.42%
50	3.041 × 10^64	3.036 × 10^64	0.17%
100	9.333 × 10^157	9.325 × 10^157	0.08%

The approximation improves as n increases. By n = 100, the error is under 0.1%.

Where Factorials Appear

Application	Formula	Why Factorials
Permutations	nPr = n!/(n-r)!	Counting ordered arrangements
Combinations	nCr = n!/(r!(n-r)!)	Counting unordered selections
Taylor series	e^x = Σ x^n/n!	Denominators control convergence
Probability	Poisson, binomial formulas	Counting possible outcomes
Gamma function	Γ(n) = (n-1)!	Extends factorials to non-integers

For applying factorials in counting problems, the permutations and combinations calculator computes nPr and nCr directly. For exponentiation, the exponent calculator handles any base to any power.

All calculations run in your browser using BigInt for exact results. No data is sent to any server.