77 Factorial — What is 77! ?
77! — Results
Exact value
=
1.4518309 × 10113 — Show full value
145183092028285869634070784086308284983740379224208358846781574688061991349156420080065207861248000000000000000000
Number of digits
=
114
Scientific notation
=
1.4518309 × 10113
Stirling's approximation
≈
1.450261 × 10113
n! ≈ √(2πn) · (n/e)ⁿ
Factorial neighbourhood of 77
The table below shows how quickly factorials grow around 77.
Calculate another factorial
Integers from 0 to 170 (171! overflows IEEE 754 double).
About 77!
77! equals 77 × 76! = 77 × 1.8854947 × 10111. In combinatorics, 77! is the number of ways to arrange 77 distinct objects in a sequence — that is, the number of permutations of a set of 77 elements.
Connections to other branches of mathematics
Like all factorials, 77! appears in the binomial coefficient C(77, k) = 77! / (k! · (77−k)!) for any 0 ≤ k ≤ 77. This counts the number of k-element subsets of a 77-element set. The row 77 of Pascal's triangle sums to 277 = 1,073,741,824….
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.