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.
No problemo. No problemo is a slang expression used in North American English to indicate that a given situation does not pose a problem. It has roughly the same meaning as the expression no problem. The expression is sometimes used as an instance of pseudo-Spanish or Mock Spanish.