I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ \displaystyle \binom{\displaystyle \frac{n}{(\log_2n)^s}}{{(\log_2n)^s}}}{\displaystyle \binom{n}{(\log_2n)^s}}$$ and $s>0$.
Could someone know something?
Are they $$\frac{n^{(1-2s)n^s}}{n^{(1-s)n^s}}=n^{-sn^s}$$ and $$(\log_2n)^{-s(\log_2n)^{s}}$$ respectively?