Question

Determinati toate valorile naturale ale nr. x pt care combinari de x luate cate 2 + aranjamente de x luate cate 2 = 30.

Daca se poate sa imi rezolvati pe o foaie, ca sa pot intelege.

Answer 1

$C_x^2+A_x^2=30\\\text{C.E. }x \geq 0\\\\\frac{x!}{2!(x-2)!}+\frac{x!}{(x-2)!}=30\\\text{C.E. }x-2 \geq 0\\\\\frac{x!}{2(x-2)!}+\frac{2x!}{2(x-2)!}=30\\\\\frac{x!+2!}{2(x-2)!}=30\\\\\frac{3x!}{2(x-2)!}=30\\\\\frac{3(x-2)!(x-1)x}{2(x-2)!}=30\\\\\frac{3x(x-1)}{2}=30\\\\3x^2-3x=60\\3x^2-3x-60=0\\x^2-x-20=0\\\Delta=1+80=81\\x_{1,2}=\frac{1\pm9}{2}\\x_1\ \textless \ 0\\x_2=5$