The factorial – basics and applications
From the definition to combinatorics and probability theory
The factorial n! (read "n factorial") is the product of all positive integers from 1 to n. The values grow extremely fast: 5! = 120, 10! = 3,628,800, 20! ≈ 2.43 × 10¹⁸ – that exceeds the number of seconds since the Big Bang. No wonder the factorial is so important in combinatorics: it counts the number of ways to arrange (permutations) n distinguishable objects.
The most important example: how many ways are there to seat 5 people at a table? The first person can sit in 5 chairs, the second in 4 remaining ones, and so on. That gives 5 × 4 × 3 × 2 × 1 = 5! = 120. The same logic applies to permutations in statistics, sequencing optimization (the travelling salesman problem) and calculating binomial coefficients: (n choose k) = n! / (k! × (n−k)!).
The special case 0! = 1 is defined by mathematical convention and follows from the recursion formula n! = n × (n−1)!: for n = 1, 1! = 1 × 0! = 1, so 0! must equal 1. It also makes intuitive sense: there is exactly one way to arrange zero elements – the empty arrangement. Without this convention, formulas like the binomial coefficient would not work for k = 0 and k = n.
In probability theory, the factorial determines how many outcomes are possible when drawing without replacement. If 6 numbers are drawn out of 49 (lottery), the number of possibilities is calculated with the binomial coefficient (49 choose 6) = 49! / (6! × 43!) ≈ 13,983,816. The huge numbers arise because factors like 49! become astronomically large but largely cancel out in the fraction. Our calculator computes factorials iteratively, which delivers precise IEEE 754 results for numbers up to 170.