Use the free Factorial Calculator below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
What Is a Factorial?
A factorial is a short way to write a long multiplication. The factorial of a number n, written n!, is the product of every whole number from n down to 1. So 5! means 5 × 4 × 3 × 2 × 1, which equals 120. By definition, 0! equals 1. That rule may look odd at first, but it keeps the formulas of counting and probability consistent, and it is the reason an empty arrangement still counts as one arrangement.
Factorials show up everywhere counting matters. They count the ways to arrange objects in a row, the ways to pick teams and committees, and the steps inside probability formulas. If you have ever wondered how many ways five books can sit on a shelf, the answer is a factorial: 5!, or 120 ways. With ten books, the number jumps to 3,628,800. That explosive growth is the most important thing to understand about factorials.
How to Use This Calculator
Enter a whole number n between 0 and 100, then press Calculate factorial. The tool shows three things: the final result, the full multiplication written out, and the running totals as the multiplication grows. For 5!, you will see 5 × 4 × 3 × 2 × 1 = 120, plus each step: 5! builds from 5, then 5 × 4 = 20, then × 3 = 60, then × 2 = 120, then × 1 = 120. Watching the running totals helps you see how fast the answer grows.
The cap at 100 exists for a good reason. 100! has 158 digits. That is far beyond what an ordinary pocket calculator can show exactly, but this tool uses exact whole-number arithmetic, so every digit it prints is correct. Past 100, the multiplication chain becomes a wall of numbers that teaches nothing extra, so the tool stops there.
A Worked Example
Suppose four friends want to sit in a row for a photo. How many seating orders are possible? The first seat can go to any of the four friends, the second seat to any of the remaining three, the third to either of the last two, and the final seat to the last person. That is 4 × 3 × 2 × 1 = 24 orders. Try 4 in the tool and you will see the full chain and the running totals match this reasoning exactly.
Now try 0. The tool returns 1. This is not a trick. There is exactly one way to arrange nothing: the empty arrangement. Mathematicians define 0! as 1 so that formulas like the combination formula keep working when you choose all of the items or none of them.
How Factorials Grow
The growth is hard to feel until you see the numbers side by side. 1! is 1, 5! is 120, 10! is 3,628,800, and 20! is already about 2.4 × 10^18, which is larger than the number of seconds since the Big Bang. This is why counting problems get impossible to list by hand very quickly, and why formulas matter more than brute force. A computer can multiply, but a person cannot list 3.6 million orders of ten books.
Factorials also appear inside other formulas. The number of ways to choose r items from n, ignoring order, divides one factorial by two others. The number of ways to arrange items when order matters divides one factorial by another. Learning factorials first makes both of those formulas feel natural rather than memorised.
Common Mistakes
The most common mistake is adding instead of multiplying: 5! is not 5 + 4 + 3 + 2 + 1 = 15. It is a product, and products grow much faster than sums. Another mistake is thinking 0! is 0. It is 1, by definition, and that definition is used in every textbook. A third mistake is applying factorials to non-whole numbers. Factorials are defined for whole numbers in this tool; the gamma function extends the idea to other numbers, but that is a different tool for a different day.
Finally, remember that a factorial answers “how many orders”, not “which order”. If you need one specific arrangement, you still have to pick it. The factorial only tells you how many choices exist in total.
Try It Yourself
Start with 3, 4 and 5 to build your feel for the pattern. Then try 10 and watch the digit count jump. Then try 20 and compare the size with 10!. That jump is the whole story of factorials: small inputs, enormous outputs, and a simple multiplication rule behind all of it. Use the tool as often as you like; every calculation runs in your browser, and nothing you type is sent anywhere.