Factorial

Definition

The Factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. By convention, $0! = 1$. $n! = n \times (n-1) \times (n-2) \times \dots \times 1$

Why It Matters

Factorials are the engine of combinatorics and probability. They represent the number of ways to arrange $n$ distinct objects and are the fundamental building blocks for calculating Permutations, Combinations, and binomial expansions.

Core Concepts

• Formula: $n! = \prod_{k=1}^{n} k$
  • How to read: “n factorial is equal to the product from k equals one to n of k.”
  • Meaning / when to use: Multiply every integer from 1 up to $n$ to find the total number of ordered arrangements.
• Recursive Property: $n! = n \times (n-1)!$
  • How to read: “n factorial is equal to n times n minus one factorial.”
  • Meaning / when to use: Useful for simplifying complex algebraic fractions in series tests and counting formulas.

Connected Concepts

• Permutations
• Combinations
• The Binomial Theorem
• Infinite Sequences