1 Factorial — What is 1! ?
1! — Results
Exact value
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1
Number of digits
=
1
Scientific notation
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1
Stirling's approximation
≈
1
Factorial neighbourhood of 1
The table below shows how quickly factorials grow around 1.
Calculate another factorial
Integers from 0 to 170 (171! overflows IEEE 754 double).
About 1!
1! = 1 is the only positive integer n for which n! = n. It marks the start of the multiplicative chain: every factorial from 2! onward is built by multiplying the previous one by n.
Connections to other branches of mathematics
Like all factorials, 1! appears in the binomial coefficient C(1, k) = 1! / (k! · (1−k)!) for any 0 ≤ k ≤ 1. This counts the number of k-element subsets of a 1-element set. The row 1 of Pascal's triangle sums to 21 = 2.
About factorials
What is a factorial?
The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1 (the empty product).
How fast do factorials grow?
Factorials grow faster than any exponential function. While 2ⁿ doubles at each step, n! multiplies by n — an ever-increasing factor. By n = 100, we already have 100! ≈ 9.33 × 10157, and 170! reaches about 7.26 × 10306.
What is Stirling's approximation?
Stirling's formula provides a practical way to estimate large factorials: n! ≈ √(2πn) · (n/e)ⁿ. The relative error is already below 1% for n = 10 and below 0.1% for n = 100. It is widely used in combinatorics, statistical mechanics, and information theory.
Factorials in combinatorics
Factorials count permutations: n! is the number of ways to arrange n distinct objects in a row. They also appear in combinations C(n,k) = n! / (k!(n−k)!), in Taylor series (the n-th term is divided by n!), and in the Gamma function: Γ(n+1) = n!.
Why does 171! overflow?
IEEE 754 double-precision floats can represent values up to ≈ 1.8 × 10308. Since 170! ≈ 7.26 × 10306 is within range but 171! ≈ 1.24 × 10309 is not, 171! evaluates to
Infinity in most languages using native floats.Frequently asked questions
0! = 1 by convention. This follows from the definition of the empty product: the product of no numbers is the multiplicative identity, 1. It also ensures that C(n,0) = n!/(0!·n!) = 1 remains consistent for all n.
100! has exactly 158 digits. This is computed as ⌊log₁₀(100!)⌋ + 1 = ⌊∑ log₁₀(k) for k=1 to 100⌋ + 1 = 157 + 1 = 158.
Exactly 52! ≈ 8.07 × 1067 ways. This number is so astronomically large that every shuffle performed in all of human history is almost certainly unique.
This tool computes exact integer values up to 170! — the largest factorial that fits in an IEEE 754 double. 170! has 307 digits.
The digit count is ⌊log₁₀(n!)⌋ + 1. By logarithm properties, log₁₀(n!) = ∑ log₁₀(k) for k=1 to n. This is computable in O(n) without big-integer arithmetic.
The Gamma function Γ(z) generalises factorials to complex numbers: Γ(n+1) = n! for any non-negative integer n. Γ(1/2) = √π, and Γ(3/2) = √π/2.