Questions: Counting Principles: Addition and Multiplication Rules

Question 1

A restaurant offers 4 appetizers, 6 entrees, and 3 desserts. How many distinct 3-course meals (one of each) are possible?

• A. 13, by adding 4 + 6 + 3
• B. 36, by multiplying only entrees and desserts
• C. 72, by multiplying 4 × 6 × 3
• D. 18, by multiplying appetizers and desserts

Answer: (C)

Choosing all three courses is an AND situation — you choose an appetizer AND an entree AND a dessert. The multiplication rule gives 4 × 6 × 3 = 72. Option A is the most common error: it applies addition, which would be correct only if you were choosing ONE course — either an appetizer OR an entree OR a dessert.

Question 2

You can travel from city A to city C either via city B (3 routes A→B, 4 routes B→C) or by a direct flight (2 options). The total number of ways to travel from A to C is 3 × 4 + 2 = 14.

• T. True
• F. False

Answer: True

This correctly combines both rules. Going via B requires choosing a route A→B AND a route B→C: multiplication gives 3 × 4 = 12. The direct flight is a separate, mutually exclusive alternative. Since you go via B OR take a direct flight, addition gives 12 + 2 = 14. The key is recognizing when to apply each rule: AND → multiply, OR (mutually exclusive) → add.

Question 3

You are creating a password of exactly 2 letters followed by 3 digits, with repetition allowed. Using the multiplication rule, how many passwords are possible?

Model answer: 26 × 26 × 10 × 10 × 10 = 676,000

Each position is filled independently: 26 choices for each of the 2 letter positions and 10 choices for each of the 3 digit positions. Because we fill all 5 positions (AND), the multiplication rule applies across all slots: 26² × 10³ = 676 × 1000 = 676,000. Independence of choices is what justifies multiplying rather than adding.