Factorial Calculator

What Is a Factorial?

The factorial of a non-negative whole number n, written n!, is the product of all positive whole numbers from 1 to n:

n! = n × (n − 1) × (n − 2) × … × 2 × 1

So 4! = 4 × 3 × 2 × 1 = 24 and 6! = 720. In the ordinary definition, factorials exist only for 0, 1, 2, 3 and so on. Negative numbers have no factorial, and decimals such as 4.5 need an extension called the gamma function, which this calculator does not compute.

How to Calculate a Factorial

1. Write down n.

2. Multiply it by n − 1, then by n − 2, and keep going down to 1.

3. The final product is n!.

For 5!: 5 × 4 = 20, 20 × 3 = 60, 60 × 2 = 120, and 120 × 1 = 120. If you already know (n − 1)!, one multiplication is enough, because n! = n × (n − 1)!: knowing 9! = 362,880 gives 10! = 10 × 362,880 = 3,628,800.

Why Is 0! Equal to 1?

It is not an arbitrary rule; two simple arguments give the same answer.

The recursive relationship: n! = n × (n − 1)!, so (n − 1)! = n! ÷ n. Going down one step at a time: 3! = 4! ÷ 4 = 6, 2! = 3! ÷ 3 = 2, 1! = 2! ÷ 2 = 1, and 0! = 1! ÷ 1 = 1.

The empty product: 0! multiplies no numbers at all. Adding nothing gives 0, the starting point for sums; multiplying nothing gives 1, the starting point for products. Counting agrees too: n! is the number of ways to arrange n objects, and there is exactly one way to arrange zero objects, the empty arrangement. That is why formulas such as nCr = n! / (r!(n − r)!) work when r = 0 or r = n.

Why Do Factorials Grow So Quickly?

Each step multiplies by a larger number instead of adding one. 5! = 120, 10! = 3,628,800, and 20! = 2,432,902,008,176,640,000, already more than 2 quintillion. Going from 10! to 20! multiplies by 11 × 12 × … × 20, which is about 670 billion.

Factorials outgrow any exponential function: from n = 25 onwards n! is larger than 10ⁿ. 70! is the first factorial above a googol (10¹⁰⁰), 100! has 158 digits, and 1,000! has 2,568. Ordinary calculators and spreadsheets use floating-point numbers, which stop being exact after 22! (they hold about 16 significant digits) and overflow to infinity at 171!, so this calculator uses exact whole-number arithmetic instead.

Trailing Zeros in a Factorial

A trailing zero comes from a factor of 10, and 10 = 2 × 5. Among the numbers 1 to n there are far more even numbers than multiples of 5, so the number of 5s decides how many 10s can be formed. Count them with

floor(n/5) + floor(n/25) + floor(n/125) + …

Every multiple of 5 gives one 5, every multiple of 25 gives an extra one, and so on. For 100!: floor(100/5) = 20 and floor(100/25) = 4, so 100! ends in 24 zeros. For 10!: floor(10/5) = 2, and indeed 10! = 3,628,800 ends in two zeros.

Number of Digits in a Factorial

5! = 120 has 3 digits, 10! = 3,628,800 has 7, and 100! has 158. For a positive number x, the digit count is floor(log₁₀ x) + 1, and log₁₀(n!) = log₁₀ 1 + log₁₀ 2 + … + log₁₀ n. For very large n, Stirling's approximation, n! ≈ √(2πn) (n/e)ⁿ, gives this sum without multiplying anything, which is how the calculator finds that 1,000,000! has 5,565,709 digits.

Where Are Factorials Used?

Counting arrangements: 5 different books can be put on a shelf in 5! = 120 orders; a deck of 52 cards can be shuffled into 52! ≈ 8.07 × 10⁶⁷ orders.

Permutations and combinations: nPr = n! / (n − r)! counts ordered selections and nCr = n! / (r!(n − r)!) counts unordered ones, the binomial coefficients in (a + b)ⁿ.

Probability: the chance that 5 shuffled books land in alphabetical order is 1 / 5! = 1/120.

Series and discrete mathematics: factorials appear in the Taylor series eˣ = 1 + x + x²/2! + x³/3! + …, in Stirling's approximation, in counting problems throughout combinatorics, and in the running time of brute-force algorithms that try every ordering.

Factorial vs Permutation vs Combination

Use a factorial, n!, when you arrange all n items: seating 6 people in a row gives 6! = 720 orders.

Use a permutation, nPr = n! / (n − r)!, when you pick r of the n items and their order matters: gold, silver and bronze among 8 runners gives 8P3 = 8 × 7 × 6 = 336.

Use a combination, nCr = n! / (r!(n − r)!), when you pick r items and order does not matter: a committee of 3 from 8 people gives 8C3 = 336 ÷ 3! = 56. A factorial is the special case nPn = n!.