Question

need answer with explanation​

Answer 1

$\large\textbf{6}$

$\cline{1-}$

$\begin{aligned}&\ ^n\!P_4=12\times\ ^n\!P_2\\\\\implies\ \ &\dfrac{n!}{(n-4)!}=12\times \dfrac{n!}{(n-2)!}\\\\\implies\ \ &\dfrac{12}{(n-3)(n-2)}=1\\\\\implies\ \ &(n-3)(n-2)=12\\\\\implies\ \ &n^2-5n-6=0\\\\\implies\ \ &n^2+n-6n-6=0\\\\\implies\ \ &n(n+1)-6(n+1)=0\\\\\implies\ \ &(n+1)(n-6)=0\\\\\implies\ \ &n=-1\quad;\quad n=6\\\\\text{But}\ \ &n>0.\quad\therefore\ \mathbf{n=6}\end{aligned}$