Question

Resolva a equação

An,2 = 1/3 . An,3

Answer 1

$\dfrac{n!}{(n-2)!}= \dfrac{1}{3}\cdot\left\{ \dfrac{n!}{(n-3)}\right\}\\\\\\ \dfrac{n\cdot(n-1)\cdot(n-2)!}{(n-2)!} = \dfrac{1}{3} \cdot\dfrac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)!}{(n-3)!}\\\\\\ n\cdot(n-1)= \dfrac{1}{3}\cdot n\cdot(n-1)\cdot(n-2)\\\\\\ \dfrac{n\cdot(n-1)}{n\cdot(n-1)}= \dfrac{1}{3}\cdot(n-2)\\\\\\ \dfrac{1}{3}\cdot(n-2)=1\\\\\\ \dfrac{1}{(n-2)}= \dfrac{1}{3}\\\\\\ 1\cdot(n-2)=1\cdot3\\ n-2=3\\ n=3+2\\\\ \Large\boxed{n=5}$