Factorial Calculator

Find the Exact Factorial of a Nonnegative Integer

Factorial Formula

Formula

\[ n! = \prod_{k=1}^{n} k = 1 \times 2 \times 3 \times \dots \times n \,, \qquad 0! = 1 \]

Where:

\(n\) — a nonnegative integer (0, 1, 2, 3, …)
\(n!\) — the factorial of \(n\), the product of all whole numbers from 1 up to \(n\)

The factorial of a number is simply every whole number from 1 to that number multiplied together. The special case \(0! = 1\) is a definition that keeps counting and algebra formulas consistent.

The number of trailing zeros of \(n!\) is given by Legendre's formula:

\[ z = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \dots \]

Factorial – Calculation Example

Example: find the factorial of 10, which is the calculator's default value.

\[ 10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \]

\[ 10! = 3{,}628{,}800 \]

Result: \(10! = 3{,}628{,}800\). It has 7 digits and 2 trailing zeros, and in scientific notation it equals \(3.6288 \times 10^{6}\).

How to Use the Factorial Calculator

Type your number into the Nonnegative integer field (any whole number from 0 to 10000; the default is 10).
Read the Exact factorial result — every digit is computed with arbitrary-precision arithmetic, so nothing is rounded.
Check the Number of digits and Trailing zeros fields for a quick sense of the size of the answer.
See the Scientific notation (exact value) line if the full number is too long to read comfortably.
Review How this result was calculated for a short breakdown of the product and the trailing-zero count.

What Is a Factorial?

The factorial of a nonnegative integer \(n\), written \(n!\), is the product of all positive integers up to \(n\). So \(5! = 1 \times 2 \times 3 \times 4 \times 5 = 120\). The notation was introduced in the early 1800s, and it has become one of the most used symbols in mathematics because factorials appear everywhere: permutations and combinations, probability distributions such as the binomial and Poisson distributions, Taylor series for \(e^x\), and many counting problems.

The most common interpretation is combinatorial: \(n!\) is the number of ways to arrange \(n\) distinct objects in a row. If you have 10 books, there are \(10! = 3{,}628{,}800\) different orders in which you can shelve them. This is why factorials show up in the binomial coefficient \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), which counts selections where order does not matter.

Factorials grow faster than any exponential function. The first few values are small — \(1! = 1\), \(5! = 120\), \(10! = 3{,}628{,}800\) — but by \(n = 20\) the result already exceeds \(2.4 \times 10^{18}\), and \(70!\) is larger than a googol. That explosive growth is exactly why this calculator uses arbitrary-precision arithmetic: ordinary floating-point numbers overflow around \(170!\), while here you get every last digit exactly, up to \(10000!\) (which has 35,660 digits).

The extra outputs help you interpret huge results. The digit count tells you the order of magnitude, the scientific notation gives a compact exact form, and the trailing-zero count answers a classic question: how many times does \(n!\) end in zero? Every trailing zero comes from a factor of \(10 = 2 \times 5\), and since factors of 5 are scarcer than factors of 2, the count is \(z = \lfloor n/5 \rfloor + \lfloor n/25 \rfloor + \lfloor n/125 \rfloor + \dots\), where the floor (integer) division drops the remainder at each step.

Assumptions, Limitations and Tips

Nonnegative integers only. Factorials are not defined for negative integers, and inputs like 4.5 are rejected. For non-integer arguments mathematicians use the gamma function, where \(\Gamma(n+1) = n!\).
Range 0 to 10000. The upper limit exists purely for performance; \(10000!\) is a 35,660-digit number, and results beyond that would take noticeably longer to compute and display.
Remember \(0! = 1\) and \(1! = 1\). These are definitions, not errors. Setting \(0! = 1\) keeps formulas like \(\binom{n}{0} = \frac{n!}{0!\,n!}\) working correctly.
The result is exact, not an approximation. Many tools return \(n!\) as a rounded float (or fail past \(170!\)); here the full integer product is computed digit by digit.
Watch for overflow elsewhere. If you feed a factorial into a calculator using standard 64-bit arithmetic, values above \(20!\) will not fit, so keep this in mind when combining results — basic operations like multiplication or division of large factorials are best done with ratio simplification instead of computing each factorial first.

Frequently Asked Questions

How do I calculate a factorial by hand?

Multiply every whole number from 1 up to n together. For example, \(6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720\). A practical trick is to build up step by step: \(2! = 2\), then \(3! = 2 \times 3 = 6\), then \(4! = 6 \times 4 = 24\), and so on — each factorial is the previous one times the next number. By hand this stays manageable up to about \(10!\); beyond that the numbers grow very quickly.

Why is 0! equal to 1 and not 0?

It is a deliberate definition, chosen so that counting formulas stay consistent. For example, there is exactly one way to arrange zero objects (the empty arrangement), and the binomial coefficient \(\binom{n}{0} = \frac{n!}{0!\,n!}\) must equal 1, which only works if \(0! = 1\). It also keeps the recursion \(n! = n \times (n-1)!\) valid at \(n = 1\). So the calculator returning 1 for \(0!\) and \(1!\) is correct, not a bug.

How many digits does n! have for large n?

You can estimate the digit count with Stirling's approximation combined with logarithms: \(\log_{10}(n!) \approx n \log_{10}(n) - n \log_{10}(e)\). The number of digits is that value plus 1, rounded down. For example, \(100!\) has 158 digits and \(1000!\) has 2,568 digits. The calculator gives you the exact digit count directly, but the formula is handy when you only need a rough sense of the size.

Can I compute factorials of negative or decimal numbers?

No — factorials are defined only for nonnegative integers. A negative integer has no factorial because the defining product cannot be reversed consistently, and a decimal like 4.5 does not fit the "product of whole numbers" pattern. For non-integer or negative arguments, mathematicians extend the idea using the gamma function, where \(\Gamma(n+1) = n!\). If you need \(\Gamma\) values, look for a dedicated gamma function calculator instead.

Why does a regular calculator give an error for factorials above 170?

Most calculators and programming languages use double-precision floating-point numbers, which max out near \(1.8 \times 10^{308}\). Since \(170! \approx 7.3 \times 10^{306}\) and \(171!\) exceeds that limit, anything above 170 overflows and returns an error or infinity. Even earlier, at \(21!\), a 64-bit integer overflows. This calculator uses arbitrary-precision arithmetic, so it computes every digit exactly up to \(10000!\).

What are the most common mistakes when working with factorials?

Typical errors include assuming \(0! = 0\) (it is 1), entering a decimal instead of a whole number, confusing \(n!\) with \(n^n\), and miscounting trailing zeros by counting factors of 2 instead of 5 — the zero count is limited by the scarcer factor, so use \(z = \lfloor n/5 \rfloor + \lfloor n/25 \rfloor + \dots\). Also avoid computing huge factorials separately when simplifying ratios like \(\frac{100!}{98!}\); cancel first and multiply what remains instead.

Where are factorials used in real life?

Factorials count arrangements, so they appear whenever order matters: scheduling, shuffling a deck of cards (52! orders), and seating plans. They are central to probability and statistics — permutations, combinations, and distributions like the binomial and Poisson all rely on them. In science and engineering, factorials appear in Taylor series expansions such as \(e^x = 1 + x + \frac{x^2}{2!} + \dots\), which underpin many numerical methods and physics formulas.