Do you need \(m\)?

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

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

Find \(n\).

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