Factorial Notation
Simple Explanation
The factorial of a positive integer n, written n!, is the product of every positive integer up to n: n! = n×(n−1)×(n−2)×...×2×1. By definition, 0! = 1.
Why Do We Need It?
Factorials show up constantly throughout this chapter — they count the number of ways to fully arrange a set of distinct objects, and they are the building block of every permutation and combination formula.
Formula
Factorial Notation
n! = n × (n−1) × (n−2) × ... × 2 × 1, with 0! = 1 by definition
The factorial of a positive integer n is the product of every positive integer up to n — it counts the number of ways to arrange n distinct objects in a row.
- n
- — a non-negative integer
When to use it: Whenever the number of full arrangements of a set of distinct objects is needed, or as a building block in permutation and combination formulas.
Worked Example
Evaluate a factorial
Evaluate 6!.
Why Does This Work?
n! directly counts the number of ways to arrange n distinct objects in a row: there are n choices for the first position, n−1 remaining choices for the second, n−2 for the third, and so on — multiplying these choices together (by the Fundamental Counting Principle) gives exactly n!.
Real-Life Example
Counting delivery route orders
A delivery driver has 6 different stops to make and wants to know how many different orders they could visit them in.
6! gives this count directly — a genuinely large number (720) that quickly grows even larger as more stops are added.
Practice
Evaluate 5! / 3!.
EasyCommon mistake
Forgetting that 0! is defined to equal 1 (not 0) — this special case comes up often in the permutation and combination formulas later in this chapter.
Quick Review
- n! = n×(n−1)×...×2×1.
- 0! = 1, by definition.
- n! counts the number of ways to arrange n distinct objects in a row.