Calculate the following limit using the Central Limit Theorem
Hint:
$$\frac{8^n}{27^n}\cdot\frac{1}{2^k}=\left( \frac{1}{3} \right)^k\cdot\left(\frac{2}{3} \right)^{3n-k}$$
thus your pmf is
$$\mathbb{P}[X=k]=\binom{3n}{k}\left( \frac{1}{3} \right)^k\cdot\left(\frac{2}{3} \right)^{3n-k}$$
that is $X\sim \text{Bin}\left(3n;\frac{1}{3} \right)$ with mean $\mathbb{E}[X]=n$ and variance $\mathbb{V}[X]=\frac{2}{3}n$