What a factorial means
A factorial counts the number of ways to arrange n distinct items in a row. With three books there are 3! = 6 possible orderings; with ten books there are already 3,628,800. That explosive growth is why factorials sit at the heart of probability and combinatorics.
By convention 0! equals 1, because there is exactly one way to arrange nothing — the empty arrangement.
Reading the result
The calculator shows the exact value, the number of digits it contains, and a compact scientific-notation estimate so you can gauge its scale at a glance.
- Exact value: every digit, computed without rounding.
- Digit count: how long the number is — 100! already has 158 digits.
- Estimate: the same value in scientific notation for quick comparison.
Where factorials show up
Factorials appear far beyond textbook exercises:
- Permutations and combinations (nPr and nCr).
- Taylor series for functions like eˣ, sin and cos.
- Probability problems involving ordered outcomes.
Limits and common mistakes
Factorials are only defined for non-negative integers, so fractional or negative inputs are rejected. To keep results responsive this tool caps n at 1000; the generalisation to non-integers is the gamma function, which is beyond its scope.
Formula
n! = n × (n−1) × … × 2 × 1, with 0! = 1Frequently asked questions
- Why is 0! equal to 1?
- There is exactly one way to arrange an empty set, and defining 0! as 1 keeps formulas for permutations and combinations consistent.