The Multiplication and Addition Principles — Solutions

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1. Q1 — 3-Course Meals

   Fluency
   • Each course is an independent choice → multiply. Total = 4 × 6 × 3 = 72 meals.

2. Q2 — Factorial Calculations

   Fluency
   • (a) 5! = 5 × 4 × 3 × 2 × 1 = 120.
   • (b) 0! = 1 (by definition).
   • (c) 8!/6! = (8 × 7 × 6!)/6! = 8 × 7 = 56.
   • (d) 10!/(2! × 8!) = (10 × 9 × 8!)/(2 × 8!) = 90/2 = 45.

3. Q3 — PIN Codes

   Fluency
   • (a) With repetition: 10⁴ = 10 000 PIN codes.
   • (b) Without repetition: 10 × 9 × 8 × 7 = 5 040 PIN codes.

4. Q4 — Coin Sequences

   Fluency
   • Each toss has 2 outcomes, independently. Total = 2⁴ = 16 sequences.

5. Q5 — Number Plates

   Understanding
   • Letters (no repeats): 26 × 25 × 24 = 15 600. Digits (no repeats): 10 × 9 × 8 = 720. Total = 15 600 × 720 = 11 232 000 plates.

6. Q6 — Committee Roles

   Understanding
   • Roles are distinct, no person holds two. President: 10, VP: 9, Secretary: 8. Total = 10 × 9 × 8 = 720 ways.

7. Q7 — Subject Selection

   Understanding
   • One from each group independently. Total = 3 × 4 × 2 = 24 combinations.

8. Q8 — Pizza Options

   Understanding
   • (a) No toppings: 3 × 2 = 6 pizzas.
   • (b) Exactly one meat: 3 × 2 × 5 = 30 pizzas.
   • (c) Exactly one topping (5 meats + 8 veg = 13 choices): 3 × 2 × 13 = 78 pizzas.

9. Q9 — 3-Digit Numbers

   Problem Solving
   • (a) Even (units 0,2,4,6,8): 9 × 10 × 5 = 450.
   • (b) Divisible by 5 (units 0 or 5): 9 × 10 × 2 = 180.
   • (c) All different digits: hundreds 9 choices (1–9), tens 9 choices (0–9 excl. hundreds), units 8 choices. Total = 9 × 9 × 8 = 648.

10. Q10 — Batting Order with Restrictions

    Problem Solving
    • Fix captain (pos 1) and vice-captain (pos 11): 1 way each. Remaining 9 players fill positions 2–10 in 9! = 362 880 batting orders.