Question

Помогите решить уравнение

Answer 1

$30C_{x-3}^{x-9}=19A_{x-4}^{4}\\\\x \in N\\\\\left \{ {{x-4\geq 4 } \atop {x>9}} \right. \Rightarrow x>9\\\\C_{x-3}^{x-9} =\frac{(x-3)!}{(x-9)!(x-3-x+9)!}=\frac{(x-3)!}{(x-9)!6!}=\frac{(x-8)(x-7)(x-6)(x-5)(x-4)(x-3)}{6!} \\\\A_{x-4}^{4} =\frac{(x-4)!}{(x-4-4)!}=\frac{(x-4)!}{(x-8)!}=(x-7)(x-6)(x-5)(x-4)\\\\30C_{x-3}^{x-9}=\frac{30(x-8)(x-7)(x-6)(x-5)(x-4)(x-3)}{6!}=\frac{30(x-8)(x-7)(x-6)(x-5)(x-4)(x-3)}{24}$

$19A_{x-4}^{4}=19(x-7)(x-6)(x-5)(x-4)\\\\\frac{(x-8)(x-7)(x-6)(x-5)(x-4)(x-3)}{24}=19(x-7)(x-6)(x-5)(x-4)\\\\(x-8)(x-3)=24*19\\\\x^{2}-3x-8x+24=456\\\\x^{2}-11x-432=0\\\\D=(-11)^{2}-4*(-432)=121+1728=1849=43^{2}\\\\x_{1}=\frac{11+43}{2}=27\\\\x_{2}=\frac{11-43}{2}=-16-neyd\\\\Otvet:\boxed{27}$