There is exactly one way to arrange an empty set — do nothing — so 0! = 1. It also keeps the rule n! = n x (n-1)! valid at n = 1 and makes combination formulas work.

Can you take the factorial of a negative number?

Not with the standard definition — factorials are defined for non-negative whole numbers. A related function called the gamma function extends the idea to many other values, but plain n! stops at 0.

How large can a factorial get before it overflows a calculator?

Many pocket calculators cap out near 69!, which is about 1.7 x 10^98. Beyond that you need software that handles very large integers, since factorials grow faster than nearly any other common function.

What is the difference between a factorial and an exponent?

An exponent like 3^4 multiplies the same number four times (81). A factorial multiplies descending numbers, so 4! = 4 x 3 x 2 x 1 = 24. Factorials use a different number at each step and grow much faster for large inputs.

What Is a Factorial? n! Explained Simply

Published 9/30/2025 · 3 min read · Everyday calculators

Lena Hoffmann — Science & education writer at Allin

Mathematics · Physics

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In short

A factorial, written n!, is the product of every whole number from n down to 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. By definition 0! = 1. Factorials count the number of ways to arrange n distinct items in order, so they sit at the heart of permutations and combinations.

A factorial multiplies every whole number down to 1. Learn what n! means, how fast it grows, and why it powers permutations and combinations.

The definition in one line

For a positive whole number n, the factorial n! equals n x (n-1) x (n-2) x ... x 2 x 1. So 4! = 4 x 3 x 2 x 1 = 24, and 6! = 720. Each factorial builds on the one before it: n! = n x (n-1)!, which is why 5! is just 5 times 4!.

The one value people trip over is 0!, which equals 1, not 0. It looks odd, but it keeps the recursive rule and the combination formulas working. There is exactly one way to arrange nothing — the empty arrangement — so counting it as 1 is the consistent choice.

How fast factorials grow

Factorials explode. 5! is 120, 10! is already 3,628,800, and 20! is about 2.43 quintillion. This growth outpaces any exponential like 2^n, because each new step multiplies by a bigger number rather than the same one. That is why arranging even a modest deck of items yields astronomically many orderings.

Why factorials matter for counting

A factorial answers a precise question: in how many orders can n distinct items be arranged? With 3 books there are 3! = 6 arrangements. Permutations of r items from n use n! / (n-r)!, and combinations — where order does not matter — use n! / (r! (n-r)!). Both formulas are built entirely from factorials.

Factorial calculatorCompute the factorial n! of a whole number.Try the tool →

Frequently asked questions

Why does 0! equal 1?: There is exactly one way to arrange an empty set — do nothing — so 0! = 1. It also keeps the rule n! = n x (n-1)! valid at n = 1 and makes combination formulas work.
Can you take the factorial of a negative number?: Not with the standard definition — factorials are defined for non-negative whole numbers. A related function called the gamma function extends the idea to many other values, but plain n! stops at 0.
How large can a factorial get before it overflows a calculator?: Many pocket calculators cap out near 69!, which is about 1.7 x 10^98. Beyond that you need software that handles very large integers, since factorials grow faster than nearly any other common function.
What is the difference between a factorial and an exponent?: An exponent like 3^4 multiplies the same number four times (81). A factorial multiplies descending numbers, so 4! = 4 x 3 x 2 x 1 = 24. Factorials use a different number at each step and grow much faster for large inputs.

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