Learning about different problem solving strategies and when to use them will give you a good start. Problem solving is a process. Most strategies provide steps that help you identify the problem and choose the best solution. There are two basic types of strategies: algorithmic and heuristic.
It should be 'No vacancies' (plural). Vacant means empty/free. Vacancy is the state of being empty, therefore 'no vacancies' = no empty rooms.
Similar term(s) : Nitric oxide (NO), nitrogen dioxide (NO 2). Definition: NO x is the generic term for a group of highly reactive gases, all of which contain nitrogen and oxygen in varying amounts [such as nitric oxide (NO) and nitrogen dioxide (NO2)].