Do you need $m$ ?

Probability Level 4

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

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 $n$ .

Do you need mmm?

Do you need $m$ ?