Sampling from a set of numbers with a fixed sum

Question

Let $s = \{x_1, x_2, \ldots, x_n\}$ be a set of $n$ random non-negative integers where $\sum_i x_i = n$. And let $\{y_1, y_2, \ldots, y_{\sqrt{n}}\}$ denote a subset of size $\sqrt{n}$ of $s$, chosen uniformly at random. Defining $y$ to be $\sum_i y_i$ I am interested in calculating the value of $y$.

By linearity of expectation, I know $E[y] = \sum_i E[y_i] = \sqrt{n}$. But can I prove with high probability that $y$ is close to its mean?

I tried using Chernoff bound but unfortunately since $x_i$'s and therefore $y_i$'s are not independent, I can't apply it here.

I also tried using Chebyshev's inequality since $y_i$'s seem to be negatively correlated but I can't calculate the variance of $y_i$ and the proof would be messy even if I do.

Does anyone have any idea for a simpler proof?