Hmm I'm not sure if I understood the question, but it looks to me like that should be two broadcast domains, because the router is the only device there that breaks up broadcast domains (the bridge and switch will segment collision domains).
However if the switches are using VLANs then that would create more broadcast domains. Though I don't think that was mentioned in the question.
It'll help if you post a link to the question you're having trouble with then we can all take a look at the diagram and all.
