Do you need mm?

(1+x)2+(1+x)3+(1+x)4++(1+x)49+(1+mx)50(1+x)^2+(1+x)^3+(1+x)^4+\cdots+(1+x)^{49} + (1+mx)^{50}

Let mm be the minimum positive integer such that the coefficient of x2x^2 in the expansion above is (3n+1)(513)(3n+1) \dbinom{51}{3}, where nn is an integer.

Find nn.

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