What is a factorial and how is it calculated?

A factorial (denoted as n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely rapidly - 10! is already 3,628,800!

Where are factorials used in real life?

Factorials are fundamental in combinatorics for calculating permutations and combinations. They're used in probability theory, statistics, algebra, and calculus (particularly in Taylor series). Real-world applications include lottery odds, tournament scheduling, password combinations, and statistical analysis.

By mathematical convention and logical consistency, 0! = 1. This definition makes many formulas work correctly, particularly in combinatorics. There's exactly one way to arrange zero objects (do nothing), so 0! = 1. Additionally, the pattern n! = n × (n-1)! requires 1! = 1 × 0!, so 0! must equal 1.

What's the difference between factorial and permutation?

A factorial (n!) counts all possible arrangements of n distinct objects. A permutation P(n,r) counts arrangements of r objects selected from n objects, calculated as n!/(n-r)!. For example, 5! = 120 arrangements of 5 objects, while P(5,3) = 60 arrangements of 3 objects chosen from 5.

How large can factorials get?

Factorials grow faster than exponential functions! 20! exceeds 2.4 quintillion, and 100! has 158 digits. Most calculators have limits (typically around 170! for floating-point representation) due to number size constraints. For very large factorials, specialized mathematical libraries are needed.

Math

Factorial Calculator

Calculate factorial of a number.

How to Use This Calculator

Enter a non-negative integer

Input the number for which you want to calculate the factorial. Most calculators accept values from 0 to around 170, though some may handle larger numbers using special algorithms.

Calculate and view result

Click calculate to see the factorial value. For smaller numbers, the calculator may show the full multiplication sequence (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120).

Understand the magnitude

Review the result and note how quickly factorials grow. The calculator may display the number in both standard and scientific notation for large values, helping you grasp the scale of the result.