Exhaustive search algorithm solving vertex cover of size $k$ in time $2^{k}n^{O(1)}$?

Question

In the wiki page of Vertex Cover, it is claimed that an exhaustive search algorithm can solve the problem(the decision version of vertex cover problem) in time $2^{k}n^{O(1)}$.

Intuitively, with a fixed parameter $k$, in the worst case, the exhaustive search needs to check whether every vertex subset of size $k$ is vertex cover. The number of vertex subset of size $k$ is $\dbinom{n}{k}$, and we have $\dbinom{n}{k}=O(n^{k})$. Meanwhile, the number of steps to check a vertex subset of size $k$ is $O(kn)$, since we only need to verify whether the number of edge connected to or contained in the vertex subset is $m$, where $m$ is the number of edge in the graph. Thus, the time complexity of the exhaustive search is $O(k n^{k+1})$.

Is there any problem with my above analysis? And how to get a exhaustive search algorithm of time complexity $2^{k}n^{O(1)}$?