If ${ }^{(n+1)}{P}_5{:}^n{P}_6=2:7,$ then the value of  n is

$\frac{{ }^{n+1}{P}_5}{{ }^n{P}_6}=\frac{2}{7}\Rightarrow 7(n+1)=2(n^2-9n+20)\Rightarrow 7n+7=2n^2-18n+40$

$\Rightarrow 2n^2-25n+33=0$

$\Rightarrow 2n^2-22n-3n+33=0\Rightarrow 2n(n-11)-3(n-11)=0$

$\Rightarrow (n-11)(2n-3)=0\Rightarrow n=11,\frac{3}{2}$

but n is a positive integer $\Rightarrow n=11.$