Factorial Calculator
n! — the product of all whole numbers from 1 to n, exactly.
Quick answer
- Formula: n! = 1 × 2 × 3 × … × n, with 0! = 1 by definition
- Worked example — n! (exact value): 120 (full example below).
Factorial Calculator
| n! (exact value) | 120 |
| Number of digits | 3 |
5! = 1 × 2 × 3 × 4 × 5 = 120 — which is why 5 different books can be arranged on a shelf in exactly 120 different orders.
The factorial of n, written n!, is the product 1 × 2 × 3 × … × n. Factorials count arrangements: 5! = 120 is the number of ways to order 5 distinct items in a line.
Factorials grow astonishingly fast — 20! already exceeds 2 quintillion, and this calculator is capped at 170!, a 307-digit number. Results are calculated exactly with big-integer arithmetic and shown in full, not rounded.
The formula
Worked example
5! = 1 × 2 × 3 × 4 × 5 = 120 — which is why 5 different books can be arranged on a shelf in exactly 120 different orders.
The example is pre-loaded in the result panel — press Calculate with your own numbers to replace it.
Frequently asked questions
Why is 0! equal to 1?
By definition, and it makes the counting formulas work: there is exactly one way to arrange zero items (do nothing). It also follows the pattern n! ÷ n = (n−1)! — 1! ÷ 1 = 1, so 0! = 1.
Where are factorials used?
In permutations and combinations (lottery odds, card shuffles), probability, and the Taylor series in calculus. A shuffled deck has 52! ≈ 8 × 10^67 possible orders — more than atoms in the galaxy.
Why stop at 170?
170! ≈ 7.26 × 10^306 is close to the largest number standard floating-point arithmetic can hold (about 1.8 × 10^308); 171! overflows it. The exact digits are still shown in full here.
Category: Math calculators
Sources & methodology
This calculator applies the formula shown on this page exactly as written, using the standard definitions for its inputs. Results are computed from the values you enter — no data is stored — and are rounded only for display.
Content last reviewed by CalcHub Editorial.
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