Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$
Solution 1:
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.
Solution 2:
Start with the identity $$\prod_{j=1}^n {n\choose j} =\prod_{k=1}^n {k^k\over k!}.$$ Since $k^k/k!\to \infty$, the same is true of its geometric mean: $$\left(\prod_{k=1}^n {k^k\over k!}\right)^{1/n}\to\infty.$$