Exact
1
Factorial of 0
Answer
0! equals 1. Zero factorial is defined as 1 — the unique empty product.
Decimal
1
Formula
0! = 1 by definition
Worked steps
1
Definition
n! is the product of all positive integers up to n. 0! is defined as 1 — the product of no numbers is 1, the multiplicative identity.
2
Calculator method
Type 0, press the x! button, press =. Result: 1.
3
Where it shows up
Factorials count the number of ways to arrange n distinct items. 0! = 1 = 1! makes the formulas for nCr and nPr work cleanly at edge cases.
About this value
The factorial of 0 is 1. 0! = 1 is a convention, not a calculation — it makes formulas like nCr(n, 0) = n!/(0!·n!) = 1 work without special cases.
Frequently asked questions
What is factorial of 0?
Factorial of 0 equals 1 as an exact value, or 1 as a decimal. The exact form is what you'd typically write in exam working; the decimal is what your calculator displays.
Why is 0! equal to 1?
By convention. The empty product (multiplying no numbers together) is 1, the multiplicative identity. This convention makes formulas like nCr(n, 0) = 1 and nPr(n, 0) = 1 hold without special cases.
How fast does factorial grow?
Very fast. 5! = 120, 10! ≈ 3.6 million, 15! ≈ 1.3 trillion. Most scientific calculators hit floating-point overflow somewhere around 170! — values bigger than that show as Infinity or scientific notation only.
Where is factorial used?
Mostly in counting problems — permutations (n!) and combinations (nCr, nPr). Also in series expansions like sin(x) = x − x³/3! + x⁵/5! − …, and in probability density functions for discrete distributions.