This is a follow-up to my previous post on Expected number of balls remaining in 1 bucket out of 2., which results in the following expression
$$ E[Y] = \sum_{y=1}^n y \binom{2n - y}{n - y} \frac{1}{2}^{2n - y} $$
I am able to solve this with Stirling's approximation in a straightforward way.
The source of the problem statement actually hinted to use CLT of the binomial distribution. But I am unsure how that applies here because the number of trials in the binomial coefficient is varying (when I've used CLT in the past to approximate binomial distributions, the number of trials is fixed). Can this problem be solved using CLT?