A factorial counts something surprisingly practical: the number of distinct ways to arrange a set of items in order. It's written with an exclamation point (5! for example), and it grows dramatically faster than most people expect — which is exactly why factorials sit at the heart of probability, permutations, and combinations.
Factorial of n (written n!) is the product of every positive integer from 1 up to n. It represents the number of unique orderings possible for n distinct items — for example, the number of different ways 5 books could be arranged on a shelf, or the number of possible finishing orders in a 5-person race.
How a Factorial Is Calculated
n! = n × (n−1) × (n−2) × ... × 2 × 1
By definition, 0! = 1 (representing the single way to arrange zero items — doing nothing at all).
A Worked Example
5! = 5 × 4 × 3 × 2 × 1 = 120 — meaning there are 120 different ways to arrange 5 distinct items in a row.
10! = 3,628,800 — over 3.6 million possible orderings for just 10 items, illustrating how quickly factorials explode in size.
20! is already over 2.4 quintillion — a number so large it has no everyday intuitive comparison, despite starting from a modest input of just 20.
Common Mistakes to Avoid
- Underestimating how fast factorials grow: unlike exponential growth, factorial growth accelerates even faster — 10! is over 3.6 million, while 15! already exceeds 1.3 trillion.
- Forgetting 0! = 1, not 0: this is a common source of errors, especially in permutation and combination formulas that rely on this specific definition.
- Confusing factorial notation with a general multiplication symbol: the exclamation point specifically means "multiply every integer down to 1," not just "multiply by itself."
- Trying to calculate large factorials without a calculator: numbers above roughly 15! or so become impractical to compute by hand — this is exactly the kind of calculation a tool is built for.
Bottom Line
Factorials count orderings, and they're the foundation of permutation and combination math. Use a Factorial Calculator to instantly calculate any factorial, and see just how quickly the results grow as the input increases.