Use this guide to understand and compute factorials. See the definition of n!, a quick reference table, fast methods for large factorials (scientific notation & Stirling’s approximation), how to find trailing zeros in n!, and how factorials power permutations and combinations.
For a non-negative integer n, the factorial n! is the product of all positive integers up to n:
0! = 1 (by definition)
1! = 1
n! = n × (n−1) × (n−2) × … × 2 × 1 for n ≥ 1
Examples:
3! = 3×2×1 = 6 ·· 5! = 5×4×3×2×1 = 120
Start at n and multiply downwards by each whole number.
Stop at 1.
If n = 0, return 1.
Tip: Factorials grow very quickly. Even 20! is a 19-digit number.
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
8! = 40,320
9! = 362,880
10! = 3,628,800
11! = 39,916,800
12! = 479,001,600
13! = 6,227,020,800
14! = 87,178,291,200
15! = 1,307,674,368,000
16! = 20,922,789,888,000
17! = 355,687,428,096,000
18! = 6,402,373,705,728,000
19! = 121,645,100,408,832,000
20! = 2,432,902,008,176,640,000
Because n! explodes in size, we often present big results as mantissa × 10^exponent.
Digits in n! (Kamenetsky/Stirling-based):
digits ≈ ⌊ n·log₁₀(n/e) + 0.5·log₁₀(2πn) ⌋ + 1
(Accurate for n ≥ 3; use 1 digit for n = 0 or 1.)
Stirling’s approximation (natural log form):
ln(n!) ≈ n·ln n − n + ½·ln(2πn)
Then: n! ≈ e^{ln(n!)} ≈ m × 10^k with
k = ⌊ ln(n!) / ln 10 ⌋ and m = 10^{ ln(n!)/ln 10 − k }.
These give reliable scientific-notation estimates (e.g., 100! ≈ 9.3326 × 10^157).
The number of trailing zeros in n! equals the number of times 10 divides n!, which is limited by the number of factor 5s (since 2s are plentiful):
zeros(n!) = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ⋯
Examples:
zeros(10!) = ⌊10/5⌋ = 2
zeros(100!) = ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24
Permutations (order matters):
P(n, r) = n! / (n−r)!
Combinations (order doesn’t matter):
C(n, r) = n! / (r! · (n−r)!)
Examples:
P(10, 3) = 10! / 7! = 720
C(10, 3) = 10! / (3!·7!) = 120
Q: What is 0! and why is it 1?
Defining 0! = 1 keeps combinatorics identities (like C(n, 0) = 1) and recurrence relations consistent.
Q: How big does n! get?
Very fast. 50! has 65 digits; 100! has 158 digits. Use scientific notation and logs for large n.
Q: Can I compute factorials for non-integers?
Yes, via the Gamma function: Γ(z) generalises factorial with n! = Γ(n+1) for integers n ≥ 0. Most calculators stick to integers for simplicity.
Q: How do I find trailing zeros without computing n!?
Use the factor-of-5 formula above—no huge products required.
Q: What’s the fastest way to estimate n!?
Use Stirling’s approximation for ln(n!), then convert to scientific notation.
Q: Why does my calculator show “Infinity” for big n?
Standard floating-point overflows. Use big-integer arithmetic (exact) or logarithms/approximations (estimate).