Factorial Calculator

Calculate n!, double factorial n!!, and permutations with step-by-step solutions.

Enter a Number

Result

10!
3,628,800
Input (n)10
Result3,628,800
Number of Digits7
Scientific Notation3.6288 x 10^6
Trailing Zeros2

Step-by-Step Multiplication

10! = 10 x 9 x 8 x ... x 2 x 1 = 3,628,800

Understanding Factorials

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. By convention, 0! = 1. Factorials grow extremely rapidly: while 10! = 3,628,800, by 20! the result exceeds 2 quintillion (2.43 × 1018). Factorials are fundamental in combinatorics, probability, and many areas of mathematics.

Factorial Variants and Related Functions

Factorial (n!)

The product of all positive integers from 1 to n. Defined as n! = n × (n-1) × ... × 1.

5! = 5 x 4 x 3 x 2 x 1 = 120

Double Factorial (n!!)

Product of all positive integers up to n with the same parity (odd or even).

7!! = 7 x 5 x 3 x 1 = 105

Permutations P(n,r)

The number of ordered arrangements of r items chosen from n distinct items.

P(n,r) = n! / (n-r)!

Combinations C(n,r)

The number of unordered selections of r items from n distinct items.

C(n,r) = n! / (r!(n-r)!)

Zero Factorial

0! = 1 by convention. This is the empty product and ensures formulas work consistently.

0! = 1

Trailing Zeros

The number of trailing zeros in n! equals the number of times 10 divides n!, which is determined by factors of 5.

Zeros = floor(n/5) + floor(n/25) + ...

Applications of Factorials

Factorials appear throughout mathematics and science: in Taylor series expansions, probability distributions (Poisson, binomial), counting permutations and combinations, calculating derangements, Stirling numbers, and in the gamma function which generalizes factorials to non-integers. They are also central to computational complexity theory.

Common Factorial Values

  • 0! = 1 (by definition)
  • 1! = 1
  • 5! = 120
  • 10! = 3,628,800
  • 15! = 1,307,674,368,000
  • 20! = 2,432,902,008,176,640,000
  • Stirling's approximation: n! ≈ sqrt(2πn) × (n/e)n