Factorials – Multiplying Down to 1
The factorial of a positive whole number n, written n!, is the product of every whole number from n down to 1: n! = n × (n − 1) × (n − 2) × … × 2 × 1. Factorials grow astonishingly fast – 5! is only 120, but 10! is already over 3.6 million.
Factorials are the essential building block for the permutation and combination formulas coming up next in this section, because they count exactly one crucial thing: the number of ways to arrange a full set of n distinct items in a row.
Computing a Factorial
n! = n × (n − 1) × … × 2 × 1. A ratio like n! ÷ m! (for m < n) simplifies to just the extra factors between m and n, without computing either factorial in full.
Evaluate 6!.
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Simplify 7! ÷ 5! without computing either factorial fully.
7! ÷ 5! = 7 × 6 (the extra factors) = 42.
Real-Life Application
- Scheduling: the number of ways to order a list of tasks is a factorial.
- Cryptanalysis: factorials describe how many possible arrangements a cipher might need to try.
- Delivery routes: the number of possible delivery orders for a set of stops is a factorial.
Key Takeaways
- n! multiplies every whole number from n down to 1.
- Factorials grow extremely quickly as n increases.
- A ratio of factorials often simplifies without full computation.
Practice: Factorials
Factorials
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