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.
Resignation. A resignation is the formal act of giving up or quitting one's office or position. A resignation can occur when a person holding a position gained by election or appointment steps down, but leaving a position upon the expiration of a term is not considered resignation. When an employee chooses to leave a position it is considered a resignation, as opposed to termination, which occurs when the employee involuntarily loses a job.