Introduction to the Fundamental Counting Principle (FCP)

Overview

• Goal: Determine the Fundamental Counting Principle and use it to solve problems.
• Some concepts are intuitive, while others are strategies.

Example: Outfits Calculation

• Scenario: A customer purchases 5 color-coordinated items:
  • Shorts: Navy, Black, Gray (3 pairs)
  • T-shirts: White, Red (2 shirts)
• Possible outfits:
  • Outfits List:
    1. Navy Shorts + White Shirt
    2. Navy Shorts + Red Shirt
    3. Black Shorts + White Shirt
    4. Black Shorts + Red Shirt
    5. Gray Shorts + White Shirt
    6. Gray Shorts + Red Shirt
• Total Outfits: 6 outfits

Tree Diagram

• Useful for organizing possible outcomes:
  • Shorts: Navy, Black, Gray
  • T-shirts: White, Red
• Shows combinations visually leading to 6 outfits.

Numerical Calculation

• Calculation: If 3 pairs of shorts and 2 t-shirts →

  3 (shorts) x 2 (t-shirts) = 6 outfits

Additional Purchases

• If the customer purchases:
  • Another pair of shorts → 4 shorts x 2 t-shirts = 8 outfits
  • Another t-shirt → 3 shorts x 3 t-shirts = 9 outfits
• Conclusion: Buying an additional t-shirt increases outfits.

Counting Methods Overview

• Sample Space: Organized listing of all possible outcomes from an experiment.
• Two common methods to organize sample space:
  1. Outcome Table
  2. Tree Diagram

Fundamental Counting Principle (FCP)

• Formula: If there are A selections of one item, B selections of a second item, C selections of a third item, then:
  • Total arrangements = A x B x C
• Applicable for any number of items.

Example: Restaurant Menu Combinations

• Menu has:
  • Meat: 2 dishes
  • Vegetable: 2 dishes
  • Rice: 3 dishes
• Using tree diagram:
  • Possible meals = 2 (meats) x 2 (vegetables) x 3 (rice) = 12 meals

Example: Combination Lock

• A lock has 40 numbers.
• To open:
  1. Rotate clockwise to first number.
  2. Rotate counterclockwise to second number.
  3. Rotate clockwise to third number.
• Repetition allowed:
  • Total combinations = 40 x 40 x 40 = 64,000 combinations
• Time to try all combinations (5 seconds each):
  • Total time = 64,000 x 5 seconds = 320,000 seconds = 88.8 hours = 3.7 days

Restricted Combinations

• If middle digit is 12:
  • Total combinations = 40 (1st) x 1 (middle) x 40 (3rd) = 1600 combinations
• If first is 28 and last is 47:
  • Total combinations = 1 (1st) x 40 (middle) x 1 (last) = 40 combinations

Permutations of Given Numbers

• For numbers 12, 28, and 47 in any order:
  • Total = 3 (1st) x 2 (2nd) x 1 (3rd) = 6 combinations

Postal Codes Example

• Format: 6 characters (odd: uppercase letters, even: digits).
• Total possibilities:
  • Letters (A-Z): 26 options
  • Digits (0-9): 10 options
  • Total = 26 x 10 x 26 x 10 x 26 x 10 = 17,576,000 postal codes
• Canada: Removes D, F, I, O, and U = 21 letters:
  • Total = 21 x 10 x 21 x 10 x 21 x 10 = 9,261,000 postal codes
• Difference: 8,315,000 fewer codes in Canada.

Conclusion

• FCP is a multiplication strategy for counting outcomes.
• Count restrictions first to ensure correct calculations.
• Use tree diagrams, outcome tables, or FCP as appropriate for different problems.