Question

Cⁿ₁₀₀=Cⁿ₉₉+C⁸⁰₉₉
sa se rezolve ecuatia

Answer 1

Cⁿ₁₀₀ = Cⁿ₉₉ + C⁸⁰₉₉

Cᵏₙ = Cᵏₙ₋₁ + Cᵏ⁻¹ₙ₋₁

(formula de recurentă a combinărilor)

=> Cⁿ₁₀₀ = Cⁿ₉₉ + Cⁿ⁻¹₉₉

=> Cⁿ⁻¹₉₉ = C⁸⁰₉₉ => n-1 = 80 => n = 81

Dar, Cⁿ⁻¹₉₉ = C⁹⁹⁻⁽ⁿ⁻¹⁾ ₉₉

=> C⁹⁹⁻⁽ⁿ⁻¹⁾ ₉₉ = C⁸⁰₉₉ => 99-(n-1) = 80 => 100-n = 80 => n = 20

=> n ∈ {20, 81}