What is 5! (Factorial of 5)?

What is a factorial?

The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. For example, 5! = 1 x 2 x 3 x 4 x 5 = 120. By convention, 0! = 1.

Quick start

Type 10 and click "Calculate n!". Result: 10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800. Scientific notation: 3.63e+6.

Factorial values from 0! to 20!

n	n!	Digits	Context
0	1	1	By definition (empty product)
1	1	1	Base case
5	120	3	Ways to arrange 5 books on a shelf
10	3,628,800	7	Ways to seat 10 people around a table
13	6,227,020,800	10	First factorial exceeding 1 billion
20	2,432,902,008,176,640,000	19	Largest factorial fitting in a 64-bit integer
52	8.07e+67	68	Ways to shuffle a deck of cards
100	9.33e+157	158	More than atoms in the observable universe
170	7.26e+306	307	Largest factorial in IEEE 754 double

Practical examples

How many ways to arrange 7 people in a queue? That is 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5,040 different orderings. Each person can stand in any position, and each arrangement is unique.
Permutations of n distinct objects = n!

Lottery: pick 6 from 49. Combinations C(49,6) = 49! / (6! x 43!) = 13,983,816. The factorial in the denominator removes the order - you don't care which ball came first. Your odds: about 1 in 14 million.
C(n,k) = n! / (k! x (n-k)!)

Shuffling a deck of 52 cards: 52! = 8.07 x 10^67 possible arrangements. That number is so large that every shuffle you have ever done - and every shuffle anyone will ever do - is almost certainly unique in the history of the universe.
52! has 68 digits - incomprehensibly large

Anagram count for "MATHEMATICS": 11 letters with repeats (M x2, A x2, T x2). Total arrangements = 11! / (2! x 2! x 2!) = 39,916,800 / 8 = 4,989,600 distinct anagrams.
Permutations with repetition: n! / (r1! x r2! x ...)

Why does 0! = 1? Convention, but with good reason. The number of ways to arrange 0 objects is 1 (there is exactly one way: do nothing). It also makes the recursion n! = n x (n-1)! work for n=1: 1! = 1 x 0! = 1 x 1 = 1.
The empty product convention: product of no factors = 1

Where factorials appear

Formula	Name	What it counts
n!	Permutations	Ways to arrange n distinct objects in a row
n! / (n-k)!	k-permutations	Ways to pick and arrange k items from n
n! / (k!(n-k)!)	Combinations C(n,k)	Ways to choose k items from n (order irrelevant)
e^x = sum x^n/n!	Taylor series	Approximating functions as infinite polynomials
e^(-m) m^k / k!	Poisson distribution	Probability of k events in a fixed interval

Stirling's approximation

For large n, computing n! exactly becomes impractical. Stirling's formula gives a close estimate:

    n! ~ sqrt(2 x pi x n) x (n/e)^n
  

For n = 10: Stirling gives 3,598,696 vs exact 3,628,800 - an error of only 0.83%. At n = 100 the relative error drops below 0.08%. This approximation is widely used in statistical mechanics and information theory.

FAQ - Frequently Asked Questions

Why does the calculator stop at 170?

JavaScript uses IEEE 754 double-precision floating point, which can represent numbers up to about 1.8 x 10^308. 170! = 7.26 x 10^306 fits, but 171! = 1.24 x 10^309 overflows to Infinity. For exact results beyond 170, you need BigInt or arbitrary-precision libraries.

Why does 0! equal 1?

Three reasons: (1) The empty product convention - a product of no factors equals 1 (the multiplicative identity). (2) It makes the recursion n! = n x (n-1)! consistent: 1! = 1 x 0! requires 0! = 1. (3) Combinatorially, there is exactly one way to arrange zero objects - do nothing.

Can you take the factorial of a fraction or negative number?

Not with the standard factorial, but the Gamma function extends factorials to all complex numbers (except negative integers). The relation is: n! = Gamma(n+1). So 0.5! = Gamma(1.5) = sqrt(pi)/2 ~ 0.886. This calculator handles non-negative integers only.

How many digits does 100! have?

158 digits. You can estimate the number of digits using Stirling's approximation: log10(n!) ~ n x log10(n/e) + 0.5 x log10(2 x pi x n). For n=100: log10(100!) ~ 157.97, so 158 digits. The exact value starts with 9332621544...

What is the difference between factorial and permutation?

n! counts all possible orderings of n distinct objects - that is a permutation of n. A k-permutation P(n,k) = n!/(n-k)! counts orderings of k objects chosen from n. When k = n, P(n,n) = n!/0! = n! - a full permutation equals the factorial.

How is factorial used in probability?

Factorials appear in almost every counting formula in probability. The binomial coefficient C(n,k) = n!/(k!(n-k)!) counts combinations. The Poisson distribution uses k! in the denominator. The multinomial coefficient generalizes to multiple categories. Without factorials, modern probability theory would not exist.

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What is 5 factorial (5!)?

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