Question

(UFRGS) A solução da equação 2Ax,4 = 4! Cx,x-5 é:
A. 15
B. 12
C. 10
D. 8
E. 6

Answer 1

$2\,.\,A_{x,4}~=~4!\,.\,C_{x,x-5}\\\\\\\\2\,.\,\dfrac{x!}{(x-4)!}~=~4.3.2.1~.~\dfrac{x!}{(x-5)!\,.\,(~x-(x-5)~)!}\\\\\\\\2\,.\,\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,(x-4)!}{(x-4)!}~=~24~.~\dfrac{x!}{(x-5)!\,.\,(~5~)!}$

$2~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,1}{1}~=~24~.~\dfrac{x!}{(x-5)!\,.\,120}$

$2~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,1}{1}~=~\dfrac{24}{120}~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,(x-4)\,.\,(x-5)!}{(x-5)!}\\\\\\\\2~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,1}{1}~=~\dfrac{24}{120}~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,(x-4)\,.\,1}{1}$

$2~.~\dfrac{120}{24}~.~\dfrac{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)\,.\,1}{x\,.\,(x-1)\,.\,(x-2)\,.\,(x-3)}~=~\dfrac{(x-4)\,.\,1}{1}\\\\\\\\\dfrac{240}{24}~=~x-4\\\\\\\\10~=~x-4\\\\\\\\x~=~10+4\\\\\\\\\boxed{x~=~14}$

Resposta:   x = 14   (não há alternativa correta)