Let $p\to\infty$, $s={\binom {p^4} p}$ and $n={\binom {p^4}{p^2}}$. Find a function $f(s)$ in the following form $$\large n=(e+o(1))^{f(s)}$$
I've tried to use the followinf asymptotics for binomial coeffs:
$$\binom n k = \frac{n^k \exp\left(k^2/2n\right)}{k!}(1+O(1))$$
And I've got the following results:
$$s={\binom {p^4} p} = \exp\left({2p^2 \ln p - \frac{1}{2} (1 + \ln 2\pi) + p^2 + \ln p + O\left(\frac{1}{p^2}\right)}\right)$$
$$n={\binom {p^4}{p^2}} = \exp\left({4p\ln p - \frac{1}{2p^2} - \left(p + \frac{1}{2}\right)\ln p + p - \frac{1}{2}\ln 2\pi + O\left(\frac{1}{p}\right)}\right)$$
What should I do next? A hint would be useful!