Factorial Calculator

Quick Answer

Last updated: 3 November 2026 ·How we calculate this: exact multiplication with big integers

How This Factorial Calculator Works

Enter a whole number n from 0 to 5,000. The calculator multiplies every whole number from 1 up to n using exact big-integer math, so no digits are lost. It shows the result, how many digits it has, how many zeros it ends in, the value in scientific notation and, for smaller n, how close Stirling's approximation comes. The full number appears in a scrolling box you can copy.

How to Use the Factorial Calculator

1. Enter a whole number from 0 to 5,000.
2. Press Calculate.
3. Read n!, its digits and trailing zeros.
4. See the scientific notation for very large values.
5. Copy the exact value from the box if you need it.

Core Formulas

• Factorial: n! = n × (n − 1) × ... × 2 × 1
• Recursive form: n! = n × (n − 1)!, with 0! = 1
• Trailing zeros: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ...
• Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ

Key Terms

• Factorial: the product of all whole numbers from 1 to n, written n!.
• Permutation: an ordered arrangement of items.
• Trailing zeros: the zeros at the end of a number.
• Stirling's approximation: a formula that estimates n! for large n.
• Gamma function: an extension of the factorial to non-whole numbers.

Worked Example

How many zeros are at the end of 100!?

• ⌊100/5⌋ = 20 multiples of 5
• ⌊100/25⌋ = 4 extra fives from 25, 50, 75 and 100
• Total = 20 + 4 = 24 trailing zeros

Real-Life Examples

• Seating: 8 people can sit in a row in 8! = 40,320 ways.
• Card shuffles: a 52-card deck can be ordered in 52! ways, a 68-digit number.
• Lottery odds: combinations, which use factorials, give the chance of winning.
• Puzzles: count the ways to arrange the letters of a word.

Factorial vs Exponent

• Factorial: multiplies decreasing numbers, n × (n − 1) × ...
• Exponent: multiplies the same number, n × n × ...
• Growth: n! grows faster than any fixed base to the n, such as 10ⁿ, for large n.

Factorials from 0 to 21

Where Factorials Are Used

• Arrangements: n! counts the ways to order n different items.
• Combinations: n! appears in nCr = n! ÷ (r!(n − r)!).
• Probability: the binomial and Poisson formulas use factorials.
• Series: e^x and sin x use factorials in their Taylor series.

Counting Arrangements With Factorials

The main everyday use of n! is counting orders. If you have n different items, there are n choices for the first place, n − 1 for the second, and so on, which multiplies out to n!.

• All items: 6 books on a shelf can be arranged in 6! = 720 ways.
• Only some items: to fill 3 places from 10 people, use 10! ÷ 7! = 720.
• Repeated items: the letters of BOOK can be arranged in 4! ÷ 2! = 12 ways, because the two Os look the same.

When order does not matter, divide again by the factorial of the group size. That gives combinations, which our Combination Calculator works out step by step. Both ideas start from the same simple factorial.

Common Mistakes to Avoid

• Thinking 0! = 0. It equals 1.
• Using decimals. The factorial here is for whole numbers. Non-whole values use the gamma function.
• Rounding big results. Normal calculators lose digits after about 15 to 17 figures.
• Counting zeros by multiples of 10 only. Each factor of 5 paired with a 2 adds a zero.
• Confusing n! with n!!. The double factorial multiplies every other number.

Frequently Asked Questions

References & Sources