$$A = \lim_{n\to\infty} \left(\binom{n}{0}\binom{n}{1}\binom{n}{2}\dots\binom{n}{n}\right)^{1/n(n+1)}$$
We need to find the value of $A$.
My attempt: I tried taking log both the sides and went from there.
$$ \ln A = \lim_{n\to\infty} \bigg(\frac1{n(n+1)} \bigg[(n+1) \ln n! - 2 \sum_{k=0}^n(n-k+1) \ln k )\bigg]\bigg)$$
I don't know how to proceed from here.
Also, can we some how utilize gamma function in this question since we will be dealing with factorials?
