Lecture Notes: Permutations and Combinations

Overview

• Topic: Understanding permutations and combinations
• Purpose: Distinguish between ordered (permutations) and unordered (combinations) arrangements of objects
• Next Steps:
  • Video 62 will include practice problems
  • Review topics from videos 60 (counting rules) and 61 (permutations and combinations)

Permutations

• Definition: Arrangements of objects where order matters
• Notation:
  • Typically written as ( _nP_k ) or ( _nP_r )
  • Represents the number of ways to arrange ( n ) objects taking ( k ) at a time
• Example:
  • For set ( S = {A, B, C} ), 2-permutations: AB, AC, BA, BC, CA, CB
  • Total of 6 permutations
• Counting Method:
  • First object: 3 choices (A, B, C)
  • Second object: 2 remaining choices
  • Total ways = 3 x 2 = 6
• Formula:
  • [ nPk = \frac{n!}{(n-k)!} ]
  • Example: ( P(3,2) = \frac{3!}{(3-2)!} = 6 )

Combinations

• Definition: Arrangements of objects where order does not matter
• Notation:
  • Typically written as ( _nC_k ) or ( _nC_r )
• Example:
  • For set ( S = {A, B, C} ), 2-combinations: AB, AC, BC
  • Total of 3 combinations
• Formula:
  • [ nCk = \frac{n!}{(n-k)!k!} ]
  • Divides permutations by ( k! ) to remove redundancies
  • Example: ( C(3,2) = \frac{3!}{(3-2)!2!} = 3 )

Practice Questions

Example 1: Poker Hands

• Problem: Number of 5-card poker hands from a 52-card deck
• Solution: Use combinations since order doesn't matter
  • Formula: ( C(52,5) = \frac{52!}{(52-5)!5!} )
  • Result: ( 2,598,960 ) different poker hands

Example 2: Marathon Runners

• Problem: Arranging 100 marathon runners in the first, second, and third positions
• Solution: Use permutations since order matters
  • Formula: ( P(100,3) = \frac{100!}{(100-3)!} )
  • Result: ( 970,200 ) possible ways

Conclusion

• Next video will focus on practice problems involving all counting rules, permutations, and combinations discussed.