[tex]{C ^{2} _{x-2}} = 21[/tex]
[tex]x \: \in \: \mathbb{N}[/tex]
[tex]x - 2 > 2 = > x > 4[/tex]
[tex] \frac{(x - 2)!}{2!(x - 2 - 2)!} = 21 [/tex]
[tex]\frac{(x - 2)!}{2(x - 4)!} = 21[/tex]
[tex] \frac{(x - 2)(x - 3)(x - 4)!}{2(x - 4)!} = 21[/tex]
[tex] \frac{(x - 2)(x - 3)}{2} = 21[/tex]
[tex](x - 2)(x - 3) = 2 \times 21[/tex]
[tex](x - 2)(x - 3) = 42[/tex]
[tex] {x}^{2} - 3x - 2x + 6 = 42[/tex]
[tex] {x}^{2} - 3x - 2x + 6 - 42=0[/tex]
[tex] {x}^{2} - 5x - 36 = 0[/tex]
[tex] {x}^{2} - 9x + 4x - 36 = 0[/tex]
[tex]x(x - 9) + 4(x - 9) = 0[/tex]
[tex](x - 9)(x + 4) = 0 [/tex]
[tex]x - 9 = 0 = > x = 9 > 0[/tex]
[tex]x + 4 = 0 = > x = - 4 < 0 \: nu \: convine[/tex]
[tex] = > x = 9[/tex]
