Let m be the smallest positive integer such that the coefficient of  $x^2$  in the expansion of  ${(1+x)}^2+{(1+x)}^3+\mathrm{...}+{(1+x)}^{49}+{(1+mx)}^{50}$  is  $(3n+1){}^{51}{C}_3$  for some positive integer n. Then the value of n is

Coefficient of  $x^2$  in the expansion of
 ${(1+x)}^2+{(1+x)}^3+\mathrm{...}+{(1+x)}^{49}+{(1+mx)}^{50}$   is
 $$\begin{gathered} { }^2{C}_2{+}^3{C}_2+\mathrm{...}{+}^{49}{C}_2{+}^{50}{C}_2{m}^2={(3n+1)}^{51}{C}_3{ \\ }^3{C}_3{+}^3{C}_2+\mathrm{...}{+}^{49}{C}_2{+}^{50}{C}_2{m}^2={(3n+1)}^{51}{C}_3{ \\ }^{50}{C}_3{+}^{50}{C}_2{m}^2={(3n+1)}^{51}{C}_3 \\ \frac{\mathrm{50.49.48}}{6}+\frac{50.49}{2}{m}^2=(3n+1)\frac{\mathrm{51.50.49}}{6} \\ {m}^2=51n+1 \end{gathered}$$
Must be a perfect square
$\Rightarrow n=5$  and  $m=16$