real analysis - Computing: $limlimits_{ntoinfty}left(prodlimits_{k=1}^{n} binom{n}{k}right)^frac{1}{n}$

I try to compute the following limit:

$$\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$$

I'm interested in finding some reasonable ways of solving the limit. I don't find any easy approach. Any hint/suggestion is very welcome.

Answer

All the binomial coefficients except the last one are at least $n$, so the $n$th root is at least $n^{\frac{n-1}{n}}$, so the limit is infinity.