Factorial Calculator
Calculate factorials of numbers
Factorial Calculator
Calculate factorial (n! = n Γ (n-1) Γ ... Γ 2 Γ 1)
Result
5! =
120
Number of Digits
3
Scientific Notation
1.2000e+2
Calculation Steps
2! = 2
3! = 6
4! = 24
5! = 120
Pro Tip: Factorial (n!) is the product of all positive integers up to n. For example, 5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120. Factorials grow extremely rapidly and are used in permutations and combinations.
Privacy & Security
Your calculations are completely private. All factorial computations are performed locally in your browser - no data is transmitted, stored, or tracked. Your mathematical work remains confidential and secure.
What is a Factorial Calculator?
A factorial calculator is a mathematical tool that computes the factorial of a non-negative integer n, denoted as n!, which equals the product of all positive integers from 1 to n. For example, 5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120. Factorials are fundamental to combinatorics, probability theory, algebra, and calculus, appearing in formulas for permutations, combinations, Taylor series, and countless mathematical and scientific applications. The factorial function grows extremely rapidly - faster than exponential functions - making even modest factorials surprisingly large: while 10! = 3,628,800, by 20! we reach 2,432,902,008,176,640,000 (over 2 quintillion), and 100! has 158 digits. This rapid growth necessitates special computational techniques for larger factorials, including arbitrary-precision arithmetic and Stirling's approximation. Factorials answer questions like "how many ways can we arrange n objects?" (n! permutations) or appear as denominators in combination formulas C(n,k) = n!/(k!(n-k)!). By definition, 0! = 1 (not 0), which makes combinatorial formulas work consistently. The factorial function extends to non-integers through the gamma function: Ξ(n) = (n-1)! for positive integers, allowing "factorials" of fractional and complex numbers. Applications span diverse fields: cryptography (calculating possible keys), genetics (DNA sequence arrangements), quality control (defect probabilities), scheduling (task orderings), game theory (strategy combinations), and statistical distributions (Poisson, binomial). This calculator handles factorials efficiently, whether you're computing 5! for homework, 20! for permutation problems, or exploring the astronomical magnitudes of larger factorials, providing instant results with educational explanations of factorial properties and applications.
Key Features
Calculate Any Factorial
Compute factorials from 0! to 1000! with arbitrary precision for large numbers
Instant Results
Real-time calculation with results displayed immediately as you enter values
Scientific Notation
Large factorials automatically displayed in scientific notation for readability
Step-by-Step Expansion
See the complete multiplication sequence: 5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120
Factorial Properties
Learn about factorial growth, Stirling's approximation, and mathematical properties
Related Calculations
Compute permutations and combinations using factorials
Large Number Support
Handle extremely large factorials with hundreds of digits accurately
Free & Educational
No registration required with detailed explanations for learning
How to Use the Factorial Calculator
Enter Your Number
Input any non-negative integer n for which you want to calculate n!. The calculator accepts values from 0 to 1000 or higher.
View the Factorial Result
See n! calculated instantly. For example, 5! = 120, 10! = 3,628,800, 20! = 2,432,902,008,176,640,000.
Review the Expansion
Examine the step-by-step multiplication showing how n! = n Γ (n-1) Γ (n-2) Γ ... Γ 3 Γ 2 Γ 1.
Check Scientific Notation
For large factorials, view the result in scientific notation: 50! β 3.04 Γ 10βΆβ΄ for easier comprehension.
Explore Related Calculations
Use the factorial to calculate permutations P(n,r) = n!/(n-r)! and combinations C(n,r) = n!/(r!(n-r)!).
Apply to Your Problem
Use the factorial result in your permutation, combination, probability, or other mathematical application.
Factorial Tips
- Remember 0! = 1: By definition, 0! equals 1, not 0. This makes combinatorial formulas work consistently.
- Factorials Grow Extremely Fast: Even 20! exceeds 2 quintillion. Beyond 100!, you're dealing with numbers having hundreds of digits.
- Use Logarithms for Large Values: When exact values aren't needed, work with ln(n!) β n ln(n) - n to avoid overflow.
- Simplify Before Calculating: For ratios like n!/(k!(n-k)!), cancel common terms first to avoid unnecessarily large intermediate values.
- Know Common Factorials: Memorize small factorials: 5!=120, 6!=720, 7!=5040, 10!=3628800 for quick reference.
- Check Integer Overflow: Standard 64-bit integers can only hold up to 20!. Use arbitrary-precision arithmetic for larger values.
Frequently Asked Questions
What is a factorial and how is it calculated?
A factorial, denoted by the exclamation mark (!), is the product of all positive integers less than or equal to a given number. For a non-negative integer n, the factorial n! equals n Γ (n-1) Γ (n-2) Γ ... Γ 3 Γ 2 Γ 1. For example, 5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120, and 7! = 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = 5,040. The factorial can be defined recursively: n! = n Γ (n-1)!, with the base case 0! = 1 and 1! = 1. This recursive definition shows how each factorial builds on the previous: 6! = 6 Γ 5! = 6 Γ 120 = 720. Factorials only apply to non-negative integers in basic mathematics, though the gamma function extends the concept to non-integers. Factorials grow extraordinarily fast - 10! = 3,628,800, but 20! exceeds 2 quintillion, and 70! has 100 digits. This rapid growth means factorials quickly exceed standard integer ranges in computing, requiring special handling. The factorial appears throughout mathematics: it counts the number of ways to arrange n distinct objects (permutations), appears in the Taylor series expansion of exponential and trigonometric functions, defines the binomial coefficients in Pascal's triangle, and is fundamental to probability calculations. Understanding factorials is essential for combinatorics, probability theory, algebra, and calculus.
Why does 0! equal 1 and not 0?
By mathematical convention and logical necessity, 0! = 1, not 0. This definition, though initially counterintuitive, makes mathematical formulas work consistently and has solid logical justification. The most intuitive explanation comes from the empty product: when multiplying zero numbers together, the result is the multiplicative identity, which is 1 (just as adding zero numbers gives 0, the additive identity). More practically, the recursive definition n! = n Γ (n-1)! requires 0! = 1 to work: if 1! = 1 Γ 0!, then 1 = 1 Γ 0!, so 0! must equal 1. In combinatorics, n! represents the number of ways to arrange n objects. There is exactly one way to arrange zero objects: do nothing. Thus 0! = 1 makes combinatorial sense. The binomial coefficient formula C(n,0) = n!/(0!(n-0)!) = n!/n! should equal 1 (there's one way to choose zero items from n items), which works only if 0! = 1. The gamma function, which extends factorials to non-integers, satisfies Ξ(1) = 0! = 1 by definition. In probability, the Poisson distribution and other formulas require 0! = 1 for mathematical consistency. If 0! were 0, many formulas would need special case exceptions, making mathematics unnecessarily complex. The definition 0! = 1 is not arbitrary but necessary for mathematical consistency, making formulas simpler and more elegant. It's one of several mathematical conventions (like 2β° = 1) that might seem odd initially but prove essential for logical coherence and formula universality.
How fast do factorials grow and why do they get so large?
Factorials grow faster than exponential functions, experiencing super-exponential growth that makes even modest factorials astronomical. To illustrate: 5! = 120, 10! = 3,628,800 (3.6 million), 15! = 1,307,674,368,000 (1.3 trillion), 20! = 2,432,902,008,176,640,000 (2.4 quintillion), 50! β 3.04 Γ 10βΆβ΄ (a number with 65 digits), 100! β 9.33 Γ 10ΒΉβ΅β· (158 digits), and 1000! has 2,568 digits. For comparison, the estimated number of atoms in the observable universe is about 10βΈβ°, exceeded by 70! Why this explosive growth? Each factorial multiplies all integers from 1 to n, so n! includes increasingly large factors. While exponential functions like 2βΏ multiply the same base repeatedly, factorials multiply ever-larger numbers: 20! = 20Γ19Γ18Γ...Γ2Γ1 involves multiplying 20 increasingly large factors. Mathematically, n! grows approximately as (n/e)βΏ Γ β(2Οn) (Stirling's approximation), showing growth faster than exponential. The ratio (n+1)!/n! = n+1 increases with n, unlike exponential functions where the ratio stays constant. This rapid growth has practical implications: factorial calculations quickly overflow standard integer types (32-bit integers can only hold up to 12!, 64-bit up to 20!), combinatorial problems involving factorials can have astronomically many solutions (52! possible card shuffles β 8 Γ 10βΆβ·), and algorithms with factorial time complexity O(n!) become impractical for n > 15-20. Understanding this growth explains why brute-force solutions checking all permutations fail for large n, why logarithms of factorials (ln n!) are often used in calculations, and why Stirling's approximation is valuable for theoretical analysis.
What is Stirling's approximation and when is it used?
Stirling's approximation is a formula for approximating factorials when exact calculation is impractical or unnecessary: n! β β(2Οn) Γ (n/e)βΏ, or in logarithmic form: ln(n!) β n ln(n) - n + Β½ln(2Οn). Named after Scottish mathematician James Stirling (18th century), this approximation becomes increasingly accurate as n grows. For n=10, Stirling gives 3,598,696 vs actual 3,628,800 (0.8% error); for n=100, error drops below 0.01%; for large n, the relative error approaches zero. The approximation is invaluable in several contexts. In statistical mechanics and thermodynamics, calculations involving factorials of huge numbers (like 10Β²Β³ particles) are impractical without approximation. In algorithm analysis, when analyzing complexity of algorithms involving factorials, Stirling's formula simplifies asymptotic analysis: n! β (n/e)βΏ shows factorial growth rate clearly. In information theory, calculating entropy and information content often requires ln(n!), where Stirling's ln(n!) β n ln(n) - n simplifies calculations. In probability theory, approximating binomial and multinomial coefficients for large n uses Stirling's formula. The approximation also extends to the gamma function for non-integer values. A simpler but less accurate version is n! β (n/e)βΏ, capturing the essence of growth without the correction factor. Understanding Stirling's approximation enables theoretical analysis of combinatorial problems, provides intuition for factorial growth rates, and allows practical calculations in physics and computer science where exact factorials are computationally infeasible or unnecessary.
How are factorials used in permutations and combinations?
Factorials are fundamental to calculating permutations (arrangements) and combinations (selections) in combinatorics. For permutations - counting ordered arrangements - the formula P(n,r) = n!/(n-r)! gives the number of ways to arrange r objects from n total objects. For example, how many ways can you arrange 3 books from 5? P(5,3) = 5!/(5-3)! = 120/2 = 60 ways. When r=n (arranging all objects), P(n,n) = n!/0! = n!, showing n! counts all possible orderings of n objects. For combinations - counting unordered selections - the formula C(n,r) = n!/(r!(n-r)!) gives the number of ways to choose r objects from n, where order doesn't matter. For example, how many ways can you choose 3 books from 5? C(5,3) = 5!/(3!Γ2!) = 120/(6Γ2) = 10 ways. Notice C(n,r) = P(n,r)/r! because each combination corresponds to r! different permutations. These formulas arise naturally: P(n,r) reflects choosing the first object (n choices), second (n-1 choices), continuing for r selections, giving nΓ(n-1)Γ...Γ(n-r+1) = n!/(n-r)!. For C(n,r), since order doesn't matter, divide by r! to eliminate counting different orderings of the same selection. Applications are ubiquitous: lottery odds (combinations of numbers), password possibilities (permutations of characters), tournament brackets (arrangements of teams), quality control sampling (combinations of items to test), genetics (DNA sequence arrangements), committee formation (combinations of people), and seating arrangements (permutations of guests). Understanding factorial-based formulas enables solving counting problems across mathematics, statistics, probability, and real-world applications.
Can you calculate factorials for negative numbers or non-integers?
In basic mathematics, factorials are defined only for non-negative integers (0, 1, 2, 3, ...), and factorials of negative numbers or non-integers don't exist in this standard definition. However, the gamma function extends the factorial concept to all complex numbers except negative integers. The gamma function Ξ(n) is defined by an integral, and for positive integers n, it satisfies Ξ(n) = (n-1)!, so Ξ(5) = 4! = 24. This shifted relationship means Ξ(n+1) = n! for non-negative integers n. For positive non-integers, the gamma function provides meaningful values: Ξ(1/2) = βΟ β 1.772, often written as (1/2)! = βΟ/2 β 0.886; Ξ(3/2) = (βΟ)/2 β 0.886, and Ξ(5/2) = (3βΟ)/4 β 1.329. These fractional factorials appear in advanced mathematics, physics, and statistics. For negative integers, the gamma function has poles (undefined values): (-1)!, (-2)!, etc., don't exist. For negative non-integers, gamma can be extended but becomes complex. In practice, most applications use integer factorials, but the gamma extension is crucial in advanced contexts: integral calculus (Gaussian integrals involve Ξ(1/2) = βΟ), probability (gamma and beta distributions use the gamma function), physics (quantum mechanics, statistical mechanics), and complex analysis. While standard calculators only handle integer factorials, mathematical software (Mathematica, MATLAB, Python's scipy) can compute gamma functions for general arguments. Understanding that factorials extend beyond integers through the gamma function reveals deeper mathematical connections but isn't necessary for typical combinatorial applications.
What are factorials used for in probability and statistics?
Factorials are essential in probability and statistics, appearing in numerous distributions, formulas, and calculations. In basic probability, factorials calculate the denominator of equally likely outcomes: the probability of a specific arrangement of n objects is 1/n!. For example, the probability of correctly guessing the order of a shuffled deck of cards is 1/52! β 1.24 Γ 10β»βΆβΈ, astronomically small. In the binomial distribution (modeling successes in n trials), the probability mass function is P(X=k) = C(n,k) Γ p^k Γ (1-p)^(n-k), where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient using factorials. This appears in coin flips, quality control, and A/B testing. The multinomial distribution extends this using factorials: P(Xβ=kβ,...,Xβ=kβ) = n!/(kβ!kβ!...kβ!) Γ pβ^kβ Γ ... Γ pβ^kβ. The Poisson distribution approximates binomial for large n: P(X=k) = (Ξ»^k Γ e^(-Ξ»))/k!, where factorials appear in the denominator. The hypergeometric distribution, used in sampling without replacement, involves multiple factorials in its formula. Taylor series expansions of probability density functions involve factorials: e^x = Ξ£(x^n/n!). In statistical inference, likelihood ratios and test statistics often involve factorial terms. In experimental design, calculating the number of possible experimental arrangements uses factorial formulas: n! possible orderings, C(n,k) possible treatment assignments. In Bayesian statistics, multinomial-Dirichlet conjugate priors involve factorial constants. In survival analysis and reliability theory, factorial moments help characterize distributions. Understanding factorials in probability enables calculating exact probabilities, working with discrete distributions, and solving complex sampling problems in statistics, quality control, and scientific research.
How do you efficiently compute large factorials in programming?
Computing large factorials in programming requires specialized techniques to handle their super-exponential growth and avoid integer overflow. For small factorials (n β€ 20 for 64-bit integers), simple iterative multiplication works: result = 1; for i from 1 to n: result *= i. This is O(n) time and O(1) space. However, 21! exceeds 64-bit range, requiring arbitrary-precision arithmetic. Languages with built-in big integers (Python, Ruby, Java's BigInteger, C++'s Boost) handle large factorials naturally: in Python, 'import math; math.factorial(100)' computes 100! exactly (158 digits). For languages without native support, use arbitrary-precision libraries (GMP for C/C++). For very large n (thousands), iterative multiplication becomes slow. Optimizations include: computing ln(n!) instead when only magnitude matters (Stirling's approximation: ln(n!) β n ln(n) - n works well), using lookup tables for small n with on-demand calculation for larger values, and prime factorization methods that represent n! as products of prime powers, storing exponents rather than the full number (more memory-efficient for theoretical work). For modular arithmetic (computing n! mod m, common in competitive programming), use (a Γ b) mod m = ((a mod m) Γ (b mod m)) mod m at each step to prevent overflow: result = 1; for i in range(1,n+1): result = (result * i) % m. Dynamic programming with memoization caches factorial values for repeated calculations. For factorial ratios like n!/(k!(n-k)!), cancel terms before multiplying to avoid unnecessary large intermediates. Understanding these techniques is crucial for competitive programming, scientific computing, cryptography (RSA uses modular factorials), and any application requiring exact large factorials or factorial-based calculations.
Why Use Our Factorial Calculator?
Calculating factorials should be instant and informative. Our factorial calculator handles any factorial from 0! to 1000! and beyond, using arbitrary-precision arithmetic for accurate results regardless of size. Whether you're solving permutation problems, calculating combinations, studying probability, or exploring factorial growth, our tool provides immediate results with step-by-step expansions and educational explanations. With support for extremely large factorials and scientific notation display, you get both precision and clarity.