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Permutations & Combinations

Overview

Ordering the chaotic.

What is it?

Counting techniques. Permutation: Order matters. Combination: Order doesn't matter.

History

Ancient Indian mathematicians used these to calculate how many ways they could combine perfumes.

Key Idea

Permutation = Position (President). Combination = Committee (Group).

Practice This Topic

Concept Guide

Plain English: If you are unlocking a safe, 1-2-3 is different from 3-2-1 (Permutation). If you are making a fruit salad, 'Apple and Banana' is the same as 'Banana and Apple' (Combination).

Real-world example: Lotteries. You usually just need to match the numbers in any order (Combination), which makes the odds slightly better than if order mattered.

How to do it

  1. Ask: Does the order change the result?
  2. Yes (Race, Password): Use Permutation Formula (nPr).
  3. No (Team, Hand of cards): Use Combination Formula (nCr).
  4. Calculate using Factorials (!).
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Common Pitfall

Calling a padlock a 'Combination Lock'. It is technically a Permutation Lock!

Word Problem
"8 runners in a race. How many ways to award Gold, Silver, Bronze?"
Reasoning: Order matters (Gold is better than Silver). We pick 3 from 8. 8 * 7 * 6 = 336 ways.

Practice Examples

Factorial
5! = 5*4*3*2*1 = 120
Multiply count down to 1.