Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is
Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is