Factorial Calculator
Calculate n! and double factorial n!! exactly using BigInt. No overflow — 1000! (2568 digits) computes instantly. Shows expanded multiplication for small n and digit count for large results.
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How It Works
Enter any non-negative integer n. The result is n × (n−1) × ... × 2 × 1. For small n (up to about 12), the expanded multiplication shows each step. The result displays with its digit count. BigInt arithmetic ensures every digit is exact — unlike standard calculators that round above 69!.
n! = n × (n−1) × (n−2) × ... × 2 × 1 | 0! = 15! = 5×4×3×2×1 = 120 | 10! = 3,628,800 | 20! ≈ 2.43×10^18Switch to the Double Factorial tab. n!! multiplies every other integer down to 1 (odd n) or 2 (even n). Examples: 9!! = 9×7×5×3×1 = 945. 10!! = 10×8×6×4×2 = 3840. Double factorial appears in the Wallis product for π/2 and in the exact values of Gaussian integrals in physics.
Odd n: n!! = n×(n−2)×...×3×1 | Even n: n!! = n×(n−2)×...×4×27!! = 7×5×3×1 = 105 | 8!! = 8×6×4×2 = 384n! grows faster than any exponential function. At n=20: 20! = 2.43 × 10^18, while 2^20 = 1,048,576. The ratio 20! / 2^20 ≈ 2.3 billion. This is why O(n!) algorithms (brute-force travelling salesman) become impractical at n=15-20, while O(2^n) algorithms remain feasible to n≈30.
n! grows as √(2πn) × (n/e)^n (Stirling's approximation)10! = 3,628,800 | 20! ≈ 2.43×10^18 | 52! ≈ 8.07×10^67Permutations P(n,r) = n!/(n−r)!: ordered selections of r items from n. Example: P(10,3) = 10!/7! = 10×9×8 = 720. Combinations C(n,r) = n!/(r!(n−r)!): unordered selections. C(10,3) = 720/6 = 120. Compute n! and (n−r)! separately using this calculator, then divide using the Big Number Calculator.
P(n,r) = n!/(n−r)! | C(n,r) = n!/(r!(n−r)!)C(52,5) = 52!/(5!×47!) = 2,598,960 (poker hands)Stirling's formula: n! ≈ √(2πn) × (n/e)^n. For n=10: √(20π) × (10/e)^10 ≈ 3,598,696 vs exact 3,628,800 — error 0.83%. For n=100: error 0.083%. The approximation improves as n grows. Useful in statistical physics and information theory where log(n!) = n×log(n) − n + ½×log(2πn) is needed.
ln(n!) ≈ n×ln(n) − n + ½×ln(2πn)100! has 158 digits | Stirling: e^(157.97) ≈ exactQuick Reference
Common factorial values — verify in the calculator above.
Definition
0!
1
Factorial
5!
120
Factorial
10!
3,628,800
Factorial
20!
2,432,902,008,176,640,000
Large n
100!
158 digits
Double !!
7!!
105
Double !!
8!!
384
Very large
1000!
2,568 digits
Tips & Shortcuts
The expanded multiplication shows for small n (up to ~12) — useful for seeing exactly how the factorial builds up step by step.
For permutations P(n,r): compute n! with this calculator, then compute (n−r)! and divide using the Big Number Calculator for exact results.
The digit count next to the result tells you the order of magnitude: 100! has 158 digits means it is between 10^157 and 10^158.
Double factorial n!! of an even number relates to regular factorial: (2n)!! = 2^n × n!. Example: 8!! = 2^4 × 4! = 16 × 24 = 384.
For very large n (above 10000), computation may take a second or two — this is normal for exact BigInt arithmetic at this scale.
Common Mistakes
Confusing n! with n^n
n! multiplies all integers from 1 to n. n^n multiplies n by itself n times. 4! = 24 but 4^4 = 256. They grow differently: n! eventually dominates n^n for large n.
Expecting a standard calculator to compute 70!
Standard calculators (including most phone calculators) overflow at 69! because 70! ≈ 1.20 × 10^100 exceeds 64-bit floating-point maximum. This BigInt calculator has no such limit.
Confusing double factorial with factorial of factorial
(n!)! is the factorial of n!, which is astronomically larger. n!! is the double factorial — only every other integer. 5! = 120, but (5!)! = 120! (a 199-digit number). 5!! = 5×3×1 = 15.
Trying to enter a decimal number for factorial
Factorial is defined for non-negative integers only. The Gamma function extends factorial to non-integers (n! = Γ(n+1)), but this calculator only handles integer inputs.
Assuming n! / (n−r)! requires computing huge intermediate factorials
P(n,r) = n×(n−1)×...×(n−r+1) — just r multiplications, not full factorial. For P(52,5) = 52×51×50×49×48 = 311,875,200. The Permutation-Combination Calculator handles this directly.
Frequently Asked Questions
n! = n × (n−1) × (n−2) × ... × 2 × 1. It counts arrangements of n distinct items. 5! = 120. 0! = 1 by convention (the empty product). Factorials grow faster than exponentials: 10! = 3,628,800 while 10^10 = 10,000,000,000. By n=20, factorial exceeds 2 quintillion (2.43 × 10^18).
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800. This is the number of distinct sequences you can form from 10 unique items — for example, the number of ways 10 runners can finish a race. The calculator shows the expanded multiplication for small n like this.
n!! multiplies every other integer. Odd n: n!! = n × (n−2) × ... × 3 × 1. Even n: n!! = n × (n−2) × ... × 4 × 2. Examples: 7!! = 7×5×3×1 = 105. 8!! = 8×6×4×2 = 384. 1!! = 1. 2!! = 2. Double factorials appear in physics integrals, combinatorics, and the Wallis product formula for π.
Three reasons: (1) Empty product: multiplying zero numbers gives 1 by convention, like how an empty sum equals 0. (2) Recurrence: n! = n × (n−1)! implies 1! = 1 × 0!, so 0! must equal 1. (3) Binomial consistency: C(n,0) = n!/(0!·n!) = 1, meaning there is exactly one way to choose nothing from a set. All three give 0! = 1.
Digit count ≈ n×log₁₀(n/e) + ½×log₁₀(2πn) by Stirling's approximation. Key values: 10! = 7 digits, 20! = 19 digits, 50! = 65 digits, 100! = 158 digits, 1000! = 2,568 digits, 10000! = 35,659 digits. The calculator shows the exact digit count alongside the full result.
Permutations (ordered arrangements): P(n,r) = n!/(n−r)!. Combinations (unordered selections): C(n,r) = n!/(r!(n−r)!). Probability of a specific ordering of n items: 1/n!. Binomial theorem coefficients. Example: a standard 52-card deck can be arranged in 52! ≈ 8.07 × 10^67 distinct orders — more arrangements than atoms in the observable universe.
No hardcoded limit. 1000! (2,568 digits) computes in about 10ms. 10000! (35,659 digits) takes 1-2 seconds. 100000! would require significant time and memory. Standard scientific calculators overflow at 69! (the result exceeds the maximum 64-bit float). This calculator uses BigInt for exact results at any size.
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