Time complexity of computing binomial coefficients (with binary representation of integers)
There is no sub-exponential algorithm. For example, $\binom{2n}{n}$ has $\log\binom{2n}{n}$ bits, and according to Stirling's approximation,

$$
\binom{2n}{n}=\Theta\left(\frac{2\sqrt{\pi n}\left(\frac{2n}{\mathrm{e}}\right)^{2n}}{\left(\sqrt{2\pi n}\left(\frac{n}{\mathrm{e}}\right)^{n}\right)^2}\right)=\Theta\left(\frac{4^n}{\sqrt{n}}\right),
$$

we have

$$
\begin{align*}
\log\binom{2n}{n}=\Omega(n).
\end{align*}
$$

So we need at least $\Omega(n)$ time to output the result of $\binom{2n}{n}$.