Calculate factorials instantly with our free online n! calculator supporting large numbers and step-by-step calculations.
Perfect for permutations, combinations, probability problems, and combinatorics with detailed mathematical explanations.
Essential tool for students, statisticians, mathematicians, and anyone working with discrete mathematics and probability theory.
Enter any non-negative whole number (0 to 1000)
Show first 15 digits for very large numbers
Display step-by-step multiplication
• Factorial of 0 is 1 (0! = 1)
• Factorial grows very quickly (10! = 3,628,800)
• Used in permutations, combinations, and probability
• Only defined for non-negative integers
Enter a non-negative integer (n) in the input field to calculate n! (n factorial), or use common factorial examples for quick reference
Choose calculation options including precision settings, display format (exact or scientific notation), and step-by-step solution display
Click 'Calculate Factorial' to instantly get the result with multiplication steps, intermediate values, and mathematical explanations
Statisticians use factorials for calculating probabilities in distributions, combinatorial analysis, and statistical modeling for research and data analysis
Computer scientists use factorials to analyze algorithm complexity, calculate permutations, and solve problems in cryptography and optimization
Physicists and chemists apply factorials in quantum mechanics, molecular arrangements, statistical mechanics, and particle physics calculations
Biologists use factorials for genetic combinations, population genetics, evolutionary calculations, and biological sequence analysis in research
Mathematicians use factorials to solve counting problems, permutations, combinations, and analyze discrete structures in abstract algebra and graph theory
Probability theorists apply factorials in calculating sample space sizes, probability distributions, and solving problems in gambling theory and risk assessment
Engineers use factorials in reliability analysis, quality control, system design, and optimization problems in mechanical and industrial engineering
Students and educators use factorial calculators for homework problems, understanding mathematical concepts, and exploring discrete mathematics in classroom settings
The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. Mathematically, n! = n × (n-1) × (n-2) × ... × 2 × 1. By definition, 0! = 1, which provides a foundation for many mathematical formulas and combinatorial calculations.
Factorials grow extremely rapidly - while 5! = 120, 10! = 3,628,800, and 20! has 19 digits. This rapid growth makes factorials essential in combinatorics where they count the number of ways to arrange or select objects. The factorial function is fundamental to permutations (arrangements) and combinations (selections).
Beyond basic counting, factorials appear in Taylor series expansions, gamma function extensions (for non-integer values), probability distributions, and many areas of advanced mathematics. Understanding factorials is crucial for discrete mathematics, probability theory, and computational complexity analysis in computer science.
Trying to calculate factorials of negative numbers
Factorials are only defined for non-negative integers. For negative inputs, our calculator shows an error. For non-integer values, you would need the gamma function, not standard factorials.
Not understanding that 0! equals 1
By mathematical convention, 0! = 1. This definition makes many formulas work consistently and represents the number of ways to arrange zero objects (which is one way - the empty arrangement).
Confusing factorials with exponential growth
Factorial growth (n!) is much faster than exponential growth (a^n). While 2^10 = 1,024, 10! = 3,628,800. This distinction is crucial in algorithm analysis and combinatorics.
Forgetting about calculator limitations for very large numbers
Factorials grow extremely quickly. While our calculator handles large numbers, results beyond certain sizes are displayed in scientific notation. For research requiring exact values, specialized software may be needed.
0! = 1 by mathematical definition. This represents the number of ways to arrange zero objects (one way - do nothing), and makes many mathematical formulas work consistently, especially in combinatorics.
Standard factorials are only defined for non-negative integers. For negative or non-integer values, you would need the gamma function, which is an extension of factorials to real and complex numbers.
Factorials grow extremely rapidly. Our calculator can handle very large numbers and displays them in scientific notation when necessary. 100! has 158 digits, while 1000! has 2,568 digits.
Permutations (nPr = n!/(n-r)!) count arrangements where order matters, while combinations (nCr = n!/(r!(n-r)!)) count selections where order doesn't matter. Both use factorials in their calculations.
Factorials calculate the size of sample spaces in probability problems. They're used in binomial probability, hypergeometric distribution, and counting favorable outcomes in equally likely scenarios.
Standard factorials only work with integers. For decimal values, you need the gamma function Γ(n+1), which extends factorials to real and complex numbers. Our calculator focuses on integer factorials.