P&C: Complete Strategy

Fundamental Principle

• Multiplication: If task A in m ways and B in n ways, both in m×n ways
• Addition: If task A in m ways OR B in n ways, total = m+n ways

Formulas

• Permutation: ⁿPᵣ = n!/(n-r)!
• Combination: ⁿCᵣ = n!/[r!(n-r)!]
• ⁿC₀ = ⁿCₙ = 1
• ⁿCᵣ = ⁿCₙ₋ᵣ

Common Patterns

• Arranging n items: n!
• Arranging n items with r identical: n!/r!
• Selecting r from n: ⁿCᵣ
• Circular arrangements: (n-1)!
• Arranging with restrictions: Fix restricted items first

Distributions

• Identical objects into distinct boxes: Stars and bars method
• Distinct objects into identical boxes: Stirling numbers

Derangements

• Number of ways no item in original position: n!/e (approximately)

CAT Strategy

• Break complex problems into simpler sub-problems
• Use complementary counting when direct counting is hard
• Always check if order matters (P vs C)