Factorial Calculator
n! = n x (n - 1) x (n - 2) x ... x 1. By definition, 0! = 1.
Enter a whole number
Enter a whole number greater than or equal to zero. Factorial is written with an exclamation mark, such as 5!. It multiplies the number by every positive whole number below it.
Use this page for counting arrangements, permutations, combinations and class examples. Factorials grow very quickly, so even modest inputs can create very large results.
What n factorial means
The factorial result is a product. 5! means 5 x 4 x 3 x 2 x 1, which equals 120. In counting problems, this often represents the number of ways to arrange items in order.
The result is not 5 multiplied by itself. That would be a power. Factorial steps downward through whole numbers until it reaches 1.
Because each new input multiplies by one more number, the results grow much faster than many people expect from small inputs.
Multiplication rule for factorials
For a positive whole number n, n! = n x (n - 1) x (n - 2) and so on down to 1. By definition, 0! = 1. That definition keeps counting formulas and recursive factorial rules consistent.
Factorial is not normally defined for negative integers in elementary calculator use. Advanced math extends related ideas with the gamma function, but that is a different topic.
Calculating 5 factorial
To calculate 6!, multiply 6 x 5 x 4 x 3 x 2 x 1. The result is 720. This can represent the number of ways to arrange 6 different items.
For 0!, the answer is 1. One way to understand it is that there is exactly one way to arrange nothing: leave the arrangement empty.
If you are checking a permutation or combination problem, write the factorial expression before multiplying it out. Seeing the expression first makes it easier to cancel shared terms and spot whether order matters.
Factorials grow very fast
A common mistake is thinking 0! should be 0 because it contains zero. Factorial is a product rule, and 0! is defined as 1 for consistency.
Another mistake is underestimating how fast factorials grow. 10! is already 3,628,800. Large inputs can become too large for ordinary display.
Also do not confuse factorial with an exclamation used in writing. In math, the symbol is an operation with a precise meaning.
Factorial Calculator FAQ
Why is 0 factorial equal to 1?
0! equals 1 by definition because it keeps factorial formulas consistent. It also matches a counting idea: there is one way to arrange zero items, the empty arrangement.
It is not saying zero times something equals one. It is defining the empty product as 1.
How do I calculate a factorial?
Multiply the number by every positive whole number below it. For 5!, calculate 5 x 4 x 3 x 2 x 1. The result is 120.
For 1!, the result is 1. For 0!, the result is also 1.
If you are using factorials in combinations or permutations, keep the factorial form until you are ready to simplify.
How can I check a factorial result?
For small inputs, expand the product by hand. For 6!, write 6 x 5 x 4 x 3 x 2 x 1. That gives 720.
For larger inputs, compare the result with the previous factorial. Since n! = n x (n - 1)!, 7! should equal 7 x 6!.
What are factorials used for?
Factorials are common in counting arrangements, permutations, combinations, probability and series formulas.
For example, 4 different books can be arranged in 4!, or 24, different orders.
They are useful whenever order matters or when a formula counts possible selections. If the word arrangement appears in a problem, factorials are often nearby.
Can factorials use decimals or negative numbers?
In basic calculator use, factorials are for whole numbers greater than or equal to zero. Decimal and negative inputs are not part of the elementary factorial definition.
Advanced math has related extensions, but they are not the same as ordinary n! calculation.
Why do factorials get large so fast?
Each step multiplies by another whole number. The product compounds quickly. For example, 6! is 720, but 10! is already 3,628,800.
This fast growth is why factorials appear in counting problems with many possible arrangements.
It also means large factorial results can be hard to read or may exceed what a basic display can show cleanly. That does not make the idea complicated, but it does make the numbers grow sharply.