Answer:
Option A
Explanation:
Coefficient of $x^{2}$ in the expansion of
{$(1+x)^{2}+(1+x)^{3}+.....+(1+x)^{49}+(1+mx)^{50}$}
$\Rightarrow$ $^{2}C_{2}+^{3}C_{2}+^{4}C_{2}+......+^{49}C_{2}+^{50}C_{2}.m^{2}$
$= (3n+1).^{51}C_{3}$
$\Rightarrow$ $^{50}C_{3}+^{50}C_{2}m^{2}=(3n+1).^{51}C_{3}$
[$\because $ $^{r}C_{r}+^{r+1}C_{r}+.....+^{n}C_{r}=^{n+1}C_{r+1}$ ]
$\Rightarrow$ $\frac{50\times 49\times48}{3\times2\times1}+\frac{50\times49}{2}\times m^{2}$
$=(3n+1)\frac{51\times 50\times 49}{3\times2\times1}$
$m^{2}=51n+1$
$\therefore$ Minimum value of $m^{2}$ for which (51n+1) is integer (perfect square) for n=5.
$\therefore$ $m^{2}=51\times5+1$
$\Rightarrow$ $m^{2}=256$
$\therefore$ m=16 and n=5
Hence, the value of n is 5
