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We assume that the inequality is given by \begin{align*} B > N C(K-1/N, T) - N C(K, T).\tag{1} \end{align*} The argument for the case with the inequality \begin{align*} B < N C(K, T) - N C(K+1/N, T) \end{align*} is similar.  For the binary option, \begin{align*} \pmb{1}_{\{S_T \ge K\}} = \begin{cases} 1, & \textrm{if } S_T \ge K,\\ 0, & ...
As I mentioned above, I am not sure what the variable $r$ is. If we ignore that, or assume the questioner wanted to say its the risk free interest rate, then it has no effect on the number of paths. Then it is clear that after 50 steps going from \$1024 to \$2500 requires a net of 4 up movements with the given $x=y^{-1}=1.25$. Thus the number of steps ...