Factorial Calculator

The factorial multiplies all whole numbers up to n.

How the Math Works

The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. Starting with 1, each successive integer is multiplied together: 1 × 2 × 3 × ... × n. By definition, 0! equals 1, representing the empty product. This creates a rapidly growing sequence where even small inputs produce large outputs, making factorials fundamental in combinatorics and probability theory.

Practical Applications

Factorials are essential for calculating permutations and combinations in statistics, probability, and discrete mathematics. When determining how many ways to arrange n distinct items, you use n!. For example, 5! = 120 tells you there are 120 ways to arrange 5 different books on a shelf. They also appear in binomial coefficients (n choose k = n!/(k!(n-k)!)), Taylor series expansions in calculus, and counting problems in computer science algorithms.

Day-to-Day Use

While you may not calculate factorials daily, they help explain why certain arrangements feel 'overwhelming.' For instance, there are 3,628,800 ways to arrange just 10 items—explaining why organizing a closet or planning a road trip with many choices can feel complex. Understanding factorials also helps grasp probability concepts in games, surveys, and decision-making, revealing why some outcomes are astronomically more likely than others when multiple factors combine.

Worked example

6! = 720.

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