Solution
= Solution
Represent $Y_n$ as a sum of $n$ independent $\operatorname{Poisson}(1)$ variables. The <central limit theorem> gives
$$
\boxed{\frac{Y_n-n}{\sqrt n}\xrightarrow{\mathrm d}N(0,1).}
$$
Since zero is a continuity point of the standard normal distribution,
$$
e^{-n}\sum_{k=0}^n\frac{n^k}{k!}
=\mathbb P(Y_n\leq n)\longrightarrow\boxed{\frac12}.
$$