On an asymptotic limit problem.

Question

Given monotone growing positive function $f(m)$ what is the slowest growing $g(m)$ we need such that $$\Big(1-\Big(1 - \Big(1-\frac1{f(m)}\Big)^\binom{m}{t}\Big)^\frac mt \Big)^{g(m)}$$ decays to $0$ as $m>0$ grows when $t=o(\log m)$ holds?