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(C) The coefficient of $x^2$ in the expansion is given by the sum of the coefficients of $x^2$ in each term:$= {^2C_2} + {^3C_2} + {^4C_2} + \cdots +

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$5$

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(C) The coefficient of $x^2$ in the expansion is given by the sum of the coefficients of $x^2$ in each term:$= {^2C_2} + {^3C_2} + {^4C_2} + \cdots + {^{49}C_2} + {^{50}C_2} \cdot m^2$Using the identity ${^nC_r} + {^nC_{r-1}} = {^{n+1}C_r}$,we know that ${^2C_2} = {^3C_3}$.Thus,the sum becomes ${^3C_3} + {^3C_2} + {^4C_2} + \cdots + {^{49}C_2} + {^{50}C_2} \cdot m^2 = {^4C_3} + {^4C_2} + \cdots + {^{49}C_2} + {^{50}C_2} \cdot m^2$Continuing this process,the sum of the first $49$ terms is ${^{50}C_3}$.So,the total coefficient is ${^{50}C_3} + {^{50}C_2} \cdot m^2$.We are given that this equals $(3n+1) \cdot {^{51}C_3}$.Note that ${^{51}C_3} = {^{50}C_3} + {^{50}C_2}$.So,${^{50}C_3} + {^{50}C_2} \cdot m^2 = (3n+1)({^{50}C_3} + {^{50}C_2})$.${^{50}C_3} + {^{50}C_2} \cdot m^2 = (3n+1){^{50}C_3} + (3n+1){^{50}C_2}$.Since ${^{50}C_3} = \frac{50-2}{3} {^{50}C_2} = \frac{48}{3} {^{50}C_2} = 16 \cdot {^{50}C_2}$,we substitute this:$16 \cdot {^{50}C_2} + {^{50}C_2} \cdot m^2 = (3n+1) \cdot 16 \cdot {^{50}C_2} + (3n+1) \cdot {^{50}C_2}$.Dividing by ${^{50}C_2}$:$16 + m^2 = 16(3n+1) + (3n+1) = 17(3n+1) = 51n + 17$.$m^2 = 51n + 1$.For $n=1$,$m^2 = 52$ (not a square).For $n=2$,$m^2 = 103$ (not a square).For $n=3$,$m^2 = 154$ (not a square).For $n=4$,$m^2 = 205$ (not a square).For $n=5$,$m^2 = 256 = 16^2$,so $m=16$.The smallest positive integer $m$ is $16$,which gives $n=5$.

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

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