Factorial Calculator
Exact integer factorials by BigInt arithmetic — no double-precision ceiling here — with Legendre trailing zeros, digit counts, and 0! = 1 argued honestly.
Factorial Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Exact integers to 500 — the page’s chosen readability horizon
- No gamma-function values between the integers
In short: 10! = 3,628,800 exactly. The anchors: 0! = 1 (the empty product — the same convention behind 0^0), 1! = 1, and the recursion (n+1)! = (n+1) × n!, so 11! = 11 × 10! = 39,916,800. Exact integers run to 500 here: 25! = 15,511,210,043,330,985,984,000,000 — 26 digits — while 100! runs 158 digits with exactly 24 trailing zeros, by Legendre: floor(100/5) + floor(100/25) = 20 + 4.
Formula
n! = n × (n−1) × × 1, 0! = 1
zeros(n!) = ∑ ⌊n/5^k⌋
the empty product is 1 — the anchor the whole factorial stands on.
Worked Example
- Multiply down. n! counts down from n to 1 — the recursion (n+1)! = (n+1) × n! is the whole engine.
- Count in exact integers. BigInt arithmetic, no float rounding: 25! is exact to its last digit, and the 170! ceiling other tools hit is a DOUBLE limit this page does not have.
- Read the structure. trailing zeros by Legendre’s sum ⌊n/5⌋ + ⌊n/25⌋ + , and the digit count as the size gauge.
52! ≈ 8 × 10^67 — the number of ways to shuffle a deck. Every shuffle ever dealt, across every table on Earth, is a rounding error against that count.
Strengths & Limits Of This Model
Where this engine is strong
- Exact BigInt arithmetic — no 170! ceiling, no float dust
- Legendre zeros printed with the running sum
Where it stops
- No permutation/combination inputs on this page
- No prime factorization view
Practical Use Cases
Counting arrangements
n! orders n distinct items — seatings, schedules, shuffles
Probability
denominators of combination counts: C(n,k) = n!/(k!(n−k)!)
Series
the exponential’s own series e^x = ∑ xⁿ/n! runs on factorials
Methodology & Editorial Standards
Refuse negatives, non-integers, and n above 500. Compute the exact integer by BigInt multiplication from 2 to n; print it grouped. Trailing zeros by Legendre’s sum of floor(n/5^k); digit count from the exact string; a scientific portrait from the double conversion when it is finite (n up to 170), with the overflow stated honestly beyond.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Factorial Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why is 0! equal to 1?
The empty product: multiplying zero numbers together gives the multiplicative identity, exactly as adding zero numbers gives 0. The recursion forces it too — 1! = 1 × 0! and 1! is 1, so 0! must be 1. The same convention is why 0^0 = 1 on the Exponent page.
Why does 170! keep appearing as a ceiling?
Because 170! is the last factorial that fits a double-precision float — 171! overflows to Infinity. That is a FLOAT limit, not a mathematics limit: this page computes in exact integers, and 500! — 1,135 digits — comes back exact to its final zero.
How are the trailing zeros counted without looking?
Legendre’s formula: every trailing zero is a factor of 10 = 2 × 5, the 2s are plentiful, so count the 5s — floor(n/5) + floor(n/25) + floor(n/125) + . For 100!: 20 + 4 = 24. The page prints the running sum so you can watch the 25s donate their extra fives.
How fast does n! grow?
Past everything. 10! already carries 7 digits; 13! passes a billion; 52! ≈ 8 × 10^67 out-counts the atoms in the galaxy. Factorials beat 2^n eventually — the Exponent page’s growth loses this race — which is why permutation counts explode and why n! appears in every ‘how many orders’ question.
What is n! used FOR?
Counting arrangements: n! lines up n distinct items — seatings, shuffles, schedules. Probability leans on it through combinations, C(n,k) = n!/(k!(n−k)!), and the exponential series e^x = ∑ xⁿ/n! runs on it — the Factorial and e are old dance partners.
Can I take a factorial of a negative or a decimal?
Not here, and not in ordinary mathematics: factorials count WHOLE arrangements. The gamma function generalizes the factorial between the integers — 0.5! is a real thing in gamma clothing — and that is a different page’s story. This page refuses negatives and non-integers by name.
How exact is exact?
Every digit is computed and printed: 25! = 15,511,210,043,330,985,984,000,000 carries all 26 digits, and 100! prints its full 158-digit integer with the scientific portrait beside it. No representation is rounded on the way to the screen.
Why does the page stop at 500?
Judgement, not arithmetic: 500! is 1,135 digits of exact integer — computable, printable, and nearly unreadable. Past that the digits outgrow the page’s honesty; the scientific portrait and Legendre counts stay meaningful far beyond where the full integer stops being worth its pixels.