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

Coefficient of x² in expansion

$=1{+}^3{\mathrm{C}}_2{+}^4{\mathrm{C}}_2{+}^5{\mathrm{C}}_2+\cdots {+}^{49}{\mathrm{C}}_2{+}^{50}{\mathrm{C}}_2\cdot {\mathrm{m}}^2 \left[{\text{ as }}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}-1}{=}^{\mathrm{n}+1}{\mathrm{C}}_{\mathrm{r}}\right]$

$\begin{array}{l}=\left({ }^3{\mathrm{C}}_3{+}^3{\mathrm{C}}_2\right){+}^4{\mathrm{C}}_2{+}^5{\mathrm{C}}_2+\cdots {+}^{49}{\mathrm{C}}_2{+}^{50}{\mathrm{C}}_2\cdot {\mathrm{m}}^2\\ =\left({ }^4{\mathrm{C}}_3{+}^4{\mathrm{C}}_2\right)+\cdots {+}^{50}{\mathrm{C}}_2{\mathrm{m}}^2\\ {=}^5{\mathrm{C}}_3+\cdots {+}^{49}{\mathrm{C}}_2{+}^{50}{\mathrm{C}}_2{\mathrm{m}}^2\\ {=}^{50}{\mathrm{C}}_3{+}^{50}{\mathrm{C}}_2{\mathrm{m}}^2{+}^{50}{\mathrm{C}}_2{-}^{50}{\mathrm{C}}_2\\ {=}^{51}{\mathrm{C}}_3{+}^{50}{\mathrm{C}}_2\left({\mathrm{m}}^2-1\right)\\ =(3\mathrm{n}+1) {51}_{C_3}(\text{ given })\end{array}$

$\therefore 3\mathrm{n}\cdot {\frac{51}{3}}^{50}{\mathrm{C}}_2{=}^{50}{\mathrm{C}}_2\left({\mathrm{m}}^2-1\right)$

$\frac{{\mathrm{m}}^2-1}{51}=\mathrm{n}$$\Rightarrow {\mathrm{m}}^2-1=51\mathrm{n}$$\Rightarrow {\mathrm{m}}^2=51\mathrm{n}+1$

n=5$\Rightarrow$${\mathrm{m}}^2=256$

value of n is 5.