Lecture Notes on Permutations and Combinations

Objective 34: Solving Problems Involving Permutations and Combinations

Introduction and Activity

• Activity: Arrange 6 volunteers in different orders to understand permutations.
• Question: How many different orders can 6 people be arranged in a line?
  • Guesses ranged from 36 to 720.
  • Correct Answer: 720 different ways, calculated using factorials.

Definitions

• Permutation: Arrangement of objects where order matters.
• Combination: Selection of objects where order does not matter.
• Example: Arranging 6 people in specific positions for a photo is a permutation.
  • Reason: Specific order is involved.

Factorials

• Definition: Product of all counting numbers from n downward to 1, denoted by n!.
  • Example: 5! = 5x4x3x2x1 = 120
• Example with 6 People: Rearranging 6 people is 6! = 720.

Calculating Factorials in a Calculator

• Use math, navigate to prb, and select the exclamation mark option.

Permutation Formula

• Formula: P(n, r) = n! / (n - r)!
• Variables:
  • n: Total number of objects.
  • r: Number of objects to arrange.
• Example: Arranging 6 people in 6 positions.
  • Calculation: 6! / (6-6)! = 720.

Example 2: Figure Skating Medals

• Scenario: 10 finalists, 3 medals (gold, silver, bronze).
• Type: Permutation (Order matters).
• Calculation: 10P3 = 10x9x8 = 720.

Combination Formula

• Formula: C(n, r) = n! / [(n - r)! x r!]
• Scenario: Choosing 2 students from 7 to present a project.
  • Type: Combination (Order doesn't matter).
  • Calculation: 7C2 = 7! / [5! x 2!] = 21.

Examples and Calculations

• Example 3: 7 shoppers in line (Permutation)
  • 7P7 = 7! = 5040
• Example 4: Selecting 4 of 13 balloons (Combination)
  • 13C4 = 13! / (9! x 4!) = 715

Calculator Shortcuts

• Permutations and Combinations:
  • Use calculator: math -> prb -> option 2 (Permutation) or 3 (Combination).
  • Enter n first, select type, then enter r.

Practice and Discussion

• Understand the difference between permutations and combinations by considering if the order is important or not.
• Use the calculator to verify the manual calculations.

Conclusion

• Key Takeaway: Understanding when to use permutations vs. combinations is crucial, as is calculating using factorials.