The Factorial Calculator computes the factorial of a non-negative integer. The factorial function is one of the most fundamental operations in mathematics, appearing in combinatorics, probability theory, algebra, and calculus.

What is a Factorial?

The factorial of a positive integer n, denoted as n!, is the product of all positive integers less than or equal to n:

Examples:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 4! = 4 × 3 × 2 × 1 = 24
• 3! = 3 × 2 × 1 = 6
• 2! = 2 × 1 = 2
• 1! = 1
• 0! = 1 (by definition)

Why is 0! = 1?

By mathematical convention and necessity, 0! is defined as 1. This definition ensures consistency in formulas, particularly in combinatorics and calculus. For instance:

• The number of ways to arrange 0 objects is 1 (the empty arrangement)
• The formula C(n,0) = n!/(0! × n!) should equal 1, requiring 0! = 1
• The pattern n!/(n-1)! = n continues smoothly: 1!/0! = 1

Recursive Definition

Factorials can also be defined recursively:

This recursive nature makes factorials efficient to compute programmatically and helps understand their mathematical properties.

Growth Rate

Factorials grow extremely rapidly. Even relatively small values produce enormous results:

• 10! = 3,628,800
• 15! = 1,307,674,368,000
• 20! = 2,432,902,008,176,640,000
• 70! ≈ 1.2 × 10¹⁰⁰ (more atoms than in the universe!)

Example 1: Basic Factorial Calculation

Calculate 7!

Example 2: Arranging Objects

Problem: How many ways can you arrange 5 books on a shelf?

Example 3: Simplifying Factorial Fractions

Simplify: 10! / 7!

Example 4: Using Factorials in Permutations

Problem: A lock has 4 different colored buttons. How many different 4-button sequences are possible?

Example 5: Double Factorial

Calculate 7!! (double factorial)

• Simplify Before Computing: When dividing factorials like 100!/98!, cancel common terms rather than computing full values. Result: 100 × 99 = 9,900.
• Use Calculator for Large Values: Beyond 12!, numbers get very large. Use a scientific calculator or software to avoid errors.
• Recognize Special Values: Memorize small factorials (0! through 10!) for quick reference in problems.
• Factorial Patterns: Notice that n! = n × (n-1)!. This helps break down complex calculations.
• Stirling's Approximation: For very large n, use n! ≈ √(2πn) × (n/e)ⁿ to approximate factorial values.
• Combinatorics Connection: Factorials are the building blocks of permutations P(n,r) = n!/(n-r)! and combinations C(n,r) = n!/(r!(n-r)!).
• Negative Numbers: Factorials are only defined for non-negative integers. (-5)! is undefined in standard mathematics.
• Gamma Function Extension: For non-integers, the gamma function Γ(n) = (n-1)! extends factorials to all real numbers.
• Computing Efficiency: In programming, use iteration or memoization instead of naive recursion for faster factorial computation.
• Trailing Zeros: Count trailing zeros in n! by summing ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ... (counts factors of 5).