🔢 Counting & Combinatorics
Count possibilities without listing them all
🔢 Counting without listing
Combinatorics is the art of counting possibilities without writing them all out. It's about being clever and systematic.
How many ways? Think about choices at each step.
🎯 The multiplication principle
If you have 3 choices for a shirt and 4 choices for trousers, you have 3 × 4 = 12 outfits.
Each independent choice multiplies the total.
🏗️ Counting with constraints
Example: Vera's tower has blocks 1, 4, 6, ?, ?, 14 (bottom to top). Each must be ≥ 2 more than the one below.
- The ? blocks go between 6 and 14
- Lower ? must be ≥ 8 (at least 6+2)
- Upper ? must be ≤ 12 (at most 14−2)
- Upper must be ≥ lower+2
- Possibilities: (8,10), (8,11), (8,12), (9,11), (9,12) → 5 ways
🔤 Arranging cards
Cards marked 1, 5, and 11 placed side by side for 4-digit numbers:
- 11 takes up 2 digit positions, 1 and 5 take 1 each → total 4 digits ✓
- Arrangements of 3 cards: 3! = 6, but some give 3-digit or 5-digit numbers
- Valid 4-digit combos: 1-5-11, 1-11-5, 5-1-11, 5-11-1, 11-1-5, 11-5-1
- Check each: 1511, 1115, 5111, 5111... wait, some produce same number → 4 different numbers
🧠 Practice Quiz
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