Factorial Calculator

Frequently Asked Questions

What is a factorial?

n! is the product of all positive integers up to n: 5! = 5×4×3×2×1 = 120. It counts the number of ways to arrange n distinct items.

Why is 0! equal to 1?

There is exactly one way to arrange zero items (the empty arrangement), and defining 0! = 1 keeps formulas like permutations and the binomial theorem consistent.

Why does the calculator stop around 170?

171! exceeds the largest number standard double-precision floating point can represent (~1.8×10³⁰⁸), so it would return Infinity. Up to 170! the magnitude is reported exactly in digits and scientific form.

How many trailing zeros does a factorial have?

Determined by how many times 5 is a factor (there are always more 2s). 100! ends in 24 zeros. The calculator counts these for you.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

The factorial of a non-negative integer n, written n!, is the product of every positive integer from 1 up to n: n! = n × (n−1) × … × 2 × 1. It is defined recursively as n! = n × (n−1)! with the base case 0! = 1, the empty product - the conventional value of multiplying no numbers, chosen so that combinatorial formulas remain consistent. The calculator multiplies iteratively from 2 up to n, then derives two extras: the number of trailing zeros via Legendre's formula, and the total digit count from ⌊log₁₀ n!⌋ + 1. IEEE-754 double precision can represent factorials exactly only up to 170!; beyond that the true value overflows, so the calculator switches to a logarithmic estimate - summing log₁₀(i) for i = 2…n - and reports the magnitude as a mantissa × a power of ten.

Worked Example: 5! and the Recursion

5! = 5 × 4 × 3 × 2 × 1 = 120. The recursive structure is visible: 6! = 6 × 5! = 6 × 120 = 720, and 7! = 7 × 720 = 5,040 - each step multiplies by the next integer, which is why growth is super-exponential. 10! = 3,628,800 (7 digits); 13! = 6,227,020,800 (already past 6 billion); 20! ≈ 2.43 × 10¹⁸. At the small end, 1! = 1 and 0! = 1 - the two inputs that share a value.

Trailing Zeros Explained

Every trailing zero of n! comes from a factor of 10 = 2 × 5 buried in the product. Even numbers contribute factors of 2 far more often than multiples of 5 contribute factors of 5, so the number of 5s is the bottleneck. Legendre's formula counts them: ⌊n÷5⌋ + ⌊n÷25⌋ + ⌊n÷125⌋ + … The higher terms catch numbers like 25 = 5² and 125 = 5³ that supply extra fives. Example: 25! has ⌊25÷5⌋ + ⌊25÷25⌋ = 5 + 1 = 6 trailing zeros; 100! has 20 + 4 + 0 = 24.

Where It Is Used

Permutations: the number of distinct orderings of n items is exactly n! - 5 books on a shelf can be arranged 120 ways.

Combinations: C(n, r) = n! ÷ (r!(n−r)!) counts unordered selections, the basis of lottery odds and the binomial coefficients.

Series: Taylor and Maclaurin expansions divide the k-th term by k!, e.g. eˣ = Σ xᵏ÷k!.

Probability: the Poisson and binomial distributions are built on factorials.

The gamma function Γ(n) = (n−1)! extends factorials to non-integers and complex numbers.

Common Mistakes & Edge Cases

Thinking 0! = 0. It equals 1 - required so C(n, 0) = 1 and the binomial theorem hold.

Taking the factorial of a negative or non-integer - undefined for ordinary factorials; the calculator rejects these and points to the gamma function for non-integers.

Expecting exact integers for large n. In 64-bit doubles even 18! already loses exact integer precision, and past 170! only the order of magnitude is reliable.

Confusing n! with n² or 2ⁿ. Factorial outpaces every fixed-base exponential: 10! = 3.6 million while 2¹⁰ = 1,024.

Reading the displayed comma-grouped value as having those zeros - trailing-zero count refers to genuine zeros at the end of n!, not formatting separators.

Frequently Asked Questions

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