Factorial Calculator

Calculate factorial (n!) of a number.

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Examples

Example 1

Example 2

Example 3

Understanding Factorial Calculations

Factorial (denoted 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). Factorials grow extremely rapidly: 10! = 3,628,800, 20! = 2,432,902,008,176,640,000. This explosive growth makes factorials impractical to compute for large n without specialized algorithms or arbitrary-precision arithmetic. Factorials appear in combinatorics (counting permutations), probability (calculating probabilities of arrangements), and calculus (Taylor series expansions).

The factorial function has a simple recursive definition: n! = n × (n-1)!, with base case 0! = 1. This translates to iterative computation: start with result = 1, multiply by each integer from 1 to n. For example, 4! computes as 1 × 1 = 1, 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24. While elegant, this naive approach fails for large n due to integer overflow. For n > 20, standard 64-bit integers overflow; BigInt or arbitrary-precision libraries are required to compute factorials accurately.

Factorials have mathematical properties useful for optimization. The ratio (n+1)!/n! = n+1 enables incremental computation. Stirling's approximation provides estimates for large factorials: n! ≈ √(2πn) × (n/e)^n. For combinatorial calculations, ratios like n!/(n-k)! simplify to n × (n-1) × ... × (n-k+1), avoiding full factorial computation. Understanding these properties enables efficient algorithms for permutations, combinations, and probability calculations without computing massive factorial values directly.

Why Factorial Calculations Matter

Combinatorics relies on factorials to count permutations and combinations. The number of ways to arrange n distinct objects is n! (permutations). For example, arranging 5 books on a shelf: 5! = 120 arrangements. Combinations (choosing k items from n) use factorials: C(n,k) = n! / (k! × (n-k)!). For example, choosing 3 people from 10: C(10,3) = 10! / (3! × 7!) = 120. Factorials provide the mathematical foundation for counting problems in statistics, probability, and discrete mathematics.

Probability calculations use factorials to determine likelihoods of specific arrangements. The probability of a specific poker hand involves factorials to count favorable outcomes and total possible outcomes. For example, the probability of a royal flush in poker: (number of royal flush hands) / C(52,5) where C(52,5) = 52!/(5! × 47!) = 2,598,960. Birthday paradox calculations, lottery odds, and cryptographic security analyses all depend on factorial-based combinatorics to quantify probabilities accurately.

Algorithm analysis and computational complexity theory use factorials to describe worst-case scenarios. Algorithms with O(n!) complexity are impractical for n > 10 due to factorial growth. For example, the traveling salesman problem has (n-1)!/2 possible routes for n cities—12 cities means 19,958,400 routes. Understanding factorial growth helps algorithm designers recognize when brute-force approaches are infeasible and motivates development of approximation algorithms, heuristics, or dynamic programming solutions.

Common Factorial Calculation Challenges

Integer overflow occurs rapidly for factorials beyond modest values. Standard 32-bit integers overflow at 13! (6,227,020,800), 64-bit integers at 21! (51,090,942,171,709,440,000). For larger factorials, use arbitrary-precision arithmetic (BigInt in JavaScript, BigInteger in Java, mpmath in Python). Floating-point approximations (using Stirling's formula or logarithms) can estimate large factorials but lose exact integer values. Always use BigInt for exact factorial computation or logarithmic methods for approximations when exact values are not required.

Negative factorials are undefined in standard mathematics. Factorial only applies to non-negative integers (0, 1, 2, ...). The gamma function generalizes factorials to real numbers: Γ(n) = (n-1)!, but this advanced topic extends beyond basic factorial computation. When implementing factorial calculators, explicitly reject negative inputs with clear error messages: 'Factorial is not defined for negative numbers'. For mathematical completeness, some systems link to gamma function calculators for real-number generalizations.

Performance degradation for large n requires optimization strategies. Computing 100! naively requires 99 multiplications of increasingly large numbers. For repeated factorial calculations (like computing C(n,k) for many k), cache results: store computed factorials in a lookup table and reuse them. For very large factorials where exact values are not needed, use logarithmic approximations: log(n!) ≈ n×log(n) - n (from Stirling's approximation). This reduces computation time from O(n) to O(1) for approximations.

Best Practices for Factorial Calculations

Use arbitrary-precision arithmetic (BigInt) for exact factorial values beyond language integer limits. In JavaScript, use BigInt: let fact = 1n; for(let i=1n; i<=n; i++) fact *= i;. This enables computing factorials up to hundreds of digits without overflow. Document the maximum factorial your system supports—BigInt enables large n but computation time and memory grow rapidly. For n > 1000, consider logarithmic methods or warn users about performance implications.

Validate input ranges before computation to prevent errors and performance issues. Reject negative numbers explicitly (factorials undefined). For very large n (> 170 for standard floats, > 1000 for BigInt in browsers), warn about computation time or impose limits. Provide clear error messages: 'Factorial requires non-negative integer input' or 'Factorial too large to compute (maximum 170 for standard precision)'. For educational tools, explain why limits exist—teaching about integer overflow and computational complexity.

Optimize with memoization or lookup tables for repeated factorial calculations. If computing many combinations C(n,k) = n!/(k!×(n-k)!), store factorials 0! through n! once and reuse them. For fixed-range applications (n ≤ 20), precompute all factorials and store in an array. This reduces O(n) computation to O(1) lookup. For combinatorial calculations, simplify ratios algebraically before computing: n!/(n-k)! = n×(n-1)×...×(n-k+1) avoids computing full factorials entirely.

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Factorial Calculator - Calculate N! Factorial

Calculate factorial (n!) of a number.

Calculate factorial (n!) of a number, which is the product of all positive integers less than or equal to n. Perfect for mathematics, combinatorics, probability calculations, and algorithm analysis. Provides step-by-step calculation and handles large factorials up to reasonable limits.

Keywords

#factorial calculator#factorial#n factorial#math calculator#combinatorics#mathematics tool

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