How could one calculate the probability that any element in $\mathbb{Z}/N\mathbb{Z}$ can be written as a subset sum of $n$ random elements in $\mathbb{Z}/N\mathbb{Z}$?

In other words, say I pick $n$ random elements $x_1,\ldots,x_n$ for $n << N$ (for example, if $N = 2^k$ then $n=k$). Then for a random element $r\in{\mathbb{Z}/N\mathbb{Z}}$, what is the probability that $r$ can be written as $\sum_i b_ix_i$, where the $b_i$ values are restricted to be either $0$ or $1$?

In theory, I expect the probability for the case of $n=k$ and $N=2^k$ to be close to $1$, since if $n=k$ there are $2^k$ choices for the sum, so if $N=2^k$ we could expect this to hit every element. The obvious problem is that there might be some collisions (for example, $x_1 = x_2 + x_5 + x_8$), and I'm running into trouble bounding the probability of these collisions (which again should be low if the $x_i$ are picked truly at random); it may be totally easy but I'm not seeing it. It would also be nice to have a more general formula for any choice of $N$ and $n$.

Anyway, any help would be great. Thanks!

P.S. I suppose one other way of thinking about this is asking, for a random subset $A\subset \mathbb{Z}/N\mathbb{Z}$ such that $|A| = n$, what is the probability that $\mathbb{Z}/N\mathbb{Z}$ can be thought of as a $\mathbb{Z}/2\mathbb{Z}$-module free on $A$. Not sure if this will make more or less sense (it seems overly complicated at least to me), but it should be the same thing.