If for some $m,n;{}^6{C}_m+2{(}^6{C}_{m+1})+{}^6{C}_{m+2}>{}^8{C}_{3}and{}^{n-1}{P}_3;{}^n{P}_4=1:8,$ then  $3{(}^n{P}_{m+1}+{}^{n+1}{C}_m)$ is equal to

${ }^6{C}_m{+}^6{C}_{m+1}{+}^6{C}_{m+1}{+}^6{C}_{m+2}>{}^8{C}_3\Rightarrow {}^{7}C_{m+1}+{}^{7}C_{m+2}>{}^{8}C_3\Rightarrow {}^{8}C_{m+2}>{}^{8}C_3$

$$\begin{gathered} \Rightarrow m+2>3, thenm=2 \Rightarrow m+2=5(\because m\ne 1) \\ \frac{{ }^{(n-1)}{P}_3}{{ }^n{P}_4}=\frac{1}{8}\Rightarrow \frac{1}{n}=\frac{1}{8}\Rightarrow n=8 \\ Now,{ }^{n+1}{C}_m{+}^n{P}_{m+1}={}^9{C}_2+{}^8{P}_3=372 \end{gathered}$$