Factorial Calculator
Compute n! for any non-negative integer.
10!
3,628,800
factorial
Digits
7
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Factorial Calculator works
The Factorial Calculator instantly computes n! — the product of all positive integers from 1 up to n — for any non-negative integer you enter. It's an essential tool for students, mathematicians, programmers, and anyone working with permutations, combinations, probability, or algorithm analysis.
At its core, the factorial function is elegantly simple: n! is defined as n × (n−1) × (n−2) × … × 2 × 1. So 5! = 5 × 4 × 3 × 2 × 1 = 120. The calculator evaluates this product instantly, which is especially valuable as n grows large — 20! already exceeds 2.4 quintillion, and manual computation becomes error-prone. The tool handles the full integer chain in microseconds and returns the exact result without rounding, making it reliable for precise combinatorics work.
A critically important special case is 0!, which is defined to equal 1 by mathematical convention. This isn't arbitrary — it ensures that combinatorial formulas like C(n, 0) = 1 remain consistent, and it aligns with the empty product principle (a product over an empty set of terms equals the multiplicative identity, 1). Many users mistakenly expect 0! to equal 0, so the calculator correctly returns 1 and this is something to keep in mind when building formulas around it.
Factorials grow at a super-exponential rate, faster than exponential functions like 2ⁿ. This explosive growth is why factorials appear throughout complexity theory — an O(n!) algorithm becomes computationally intractable very quickly. For example, 10! = 3,628,800 while 20! ≈ 2.43 × 10¹⁸. Because standard 64-bit integers overflow around 20!, calculators often need arbitrary-precision arithmetic for larger values. This calculator supports large inputs and returns exact integer results, not floating-point approximations.
The most common real-world applications of n! are in permutations and combinations. The number of ways to arrange n distinct objects in a sequence is exactly n! — this is a permutation of n items. When you want the number of ways to choose k items from n without regard to order, the binomial coefficient C(n, k) = n! / (k! × (n−k)!) uses factorial in all three places. Misapplying factorial — for instance, forgetting to divide by k! when order doesn't matter — is one of the most frequent mistakes in introductory combinatorics, and understanding exactly what n! counts helps prevent it.
Formula
n! = n × (n−1) × … × 2 × 1
Pro tips
- Remember that 0! = 1, not 0 — this trips up many students and will silently break probability formulas if assumed otherwise.
- When computing combinations C(n, k), always simplify n! / (n−k)! by cancellation before multiplying — for example, 10! / 7! = 10 × 9 × 8, not the full product — this avoids unnecessarily large intermediate numbers.
- For very large n (say, n > 100), consider whether you actually need the exact integer value or just its logarithm; log(n!) = Σ log(k) for k=1 to n, which is far more manageable in scientific and statistical work.
- In coding and algorithm analysis, recognize that any algorithm requiring you to enumerate all n! permutations is feasible only up to roughly n = 12 in practice — use this calculator to quickly gauge feasibility before implementing.
- Double-check whether your problem involves ordered or unordered selections before applying factorial: use P(n, k) when order matters (passwords, race finishes) and C(n, k) when it doesn't (lottery picks, committee selection).
Key terms
- Factorial (n!)
- — The product of all positive integers from 1 to n, with 0! defined as 1 by convention.
- Permutation
- — An ordered arrangement of items; the number of ways to arrange n distinct items taken k at a time is P(n, k) = n! / (n−k)!.
- Combination
- — An unordered selection of items; C(n, k) = n! / (k! × (n−k)!) counts the ways to choose k items from n without regard to order.
- Combinatorics
- — The branch of mathematics concerned with counting, arrangement, and selection of objects, in which factorial is a foundational operation.
- Binomial Coefficient
- — The value C(n, k), read 'n choose k,' representing the number of k-element subsets of an n-element set, computed using factorials.
- Arbitrary-Precision Arithmetic
- — A computation method that handles integers of any size exactly, necessary for factorials beyond 20! which exceed standard 64-bit integer limits.
