What are the components of binary strings? $C_{9}$ is the graph with vertices representing all binary strings of length nine. Two strings are adjacent if and only if they differ in exactly three positions. How can I compute how many components $C_{9}$ has?

I think the answer should be $1$, as it should be possible to reach all vertices from any vertex. For example, the vertex $100000000$ is linked to $000000000$ by the path
$$000000000\to 011100000\to 010010000\to 100000000$$
My intuition tells me that such paths should be possible for any vertex. At the very least, it's obvious that it's possible to link all vertices with one $1$ to the vertex $000000000$, and similarly, you can link all vertices with one zero to $111111111$.
I think a similar process could be done to "add" another one, i.e. to link a vertex with one "1" to one with two of them.
