Permutations and Combinations: One-Page Formula Sheet

Permutations and Combinations — formula sheet

Formulas

• ⁿPᵣ = n!/(n−r)!, ⁿCᵣ = n!/(r!(n−r)!)
• Arrangements with repeats: n!/(p!q!…)
• Non-negative integer solutions of x₁+…+xᵣ = n: ⁿ⁺ʳ⁻¹Cᵣ₋₁
• Derangements Dₙ = n!(1 − 1/1! + 1/2! − … )

Key ideas

• Order matters → permutation; selection only → combination
• Identical objects reduce arrangements by the factorial of their count
• Circular arrangements of n distinct objects: (n − 1)!
• Use complement counting for 'at least one'

Traps to avoid

• Double-counting cases
• Treating identical objects as distinct
• Forgetting necklace arrangements divide by 2

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