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 n P m +…

Given: ${}^6{C}_m+2\left({}^6{C}_{m+1}\right){+}^6{C}_{m+2}{>}^8{C}_3$

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

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

${\Rightarrow }^8{C}_{m+2}{>}^8{C}_3$

$\Rightarrow m=2$

Also, ${}^{n-1}{P}_3{:}^n{P}_4=1:8$

$\Rightarrow \frac{\left(n-1\right)\left(n-2\right)\left(n-3\right)}{n\left(n-1\right)\left(n-2\right)\left(n-3\right)}=\frac{1}{8}$

$\Rightarrow n=8$

${\Rightarrow }^n{P}_{m+1}{+}^{n+1}{C}_m{=}^8{P}_3{+}^9{C}_2$

${\Rightarrow }^n{P}_{m+1}{+}^{n+1}{C}_m=8\times 7\times 6+\frac{9\times 8}{2}$

${\Rightarrow }^n{P}_{m+1}{+}^{n+1}{C}_m=372$