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.
At this point, the accused is expected to enter a plea: usually guilty, not guilty, or no contest. The no-contest plea means that the accused is not admitting guilt but will not contest the charges. Parties can waive an arraignment and merely enter a plea.