Question

Scrieti si rezolvarile va rog .

Answer 1

[tex]a)\:C_{x-2}^2=21\\\\\frac{(x-2)!}{2!(x-4)!}=21\\ \frac{(x-4)!(x-3)(x-2)}{2(x-4)!}=21\\ \frac{(x-3)(x-2)}{2}=21\\(x-3)(x-2)=42\\x^2-2x-3x+6-42=0\\x^2-5x-36=0\\\Delta=25+144=169\\x_{1,2}=\frac{5\pm13}{2}\\x_1\ \textless \ 0\\x_2=9[/tex]

$b)\:C_{x+2}^2=15\\\\\frac{(x+2)!}{2!(x+2-2)!}=15\\\\\frac{(x+2)!}{2x!}=15\\\\\frac{x!(x+1)(x+2)}{2x!}=15\\(x+1)(x+2)=30\\x^2+2x+x+2-30=0\\x^2+3x-18=0\\\Delta=9+72=81\\x_{1,2}=\frac{-3\pm9}{2}\\x_1\ \textless \ 0\\x_2=3$

$d)\:C_x^{x-3}=2C_x^{x-2}\\\\\frac{x!}{(x-3)!(x-x+3)!}=2\frac{x!}{(x-2)!(x-x+2)!}\\\\\frac{(x-3)!(x-2)(x-1)x}{(x-3)3!}=2\frac{(x-2)!(x-1)x}{(x-2)!2!}\\\\\frac{(x-2)(x-1)x}{6}=x(x-1)\\x(x-1)(x-2)=6x(x-1)\\x-2=6\leftrightarrow x=8$