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.
Difference between the vacated position and the vacant position. 'Vacate' means to leave a previously occupied place, for example, a seat or a house. 'Vacant' means a place that isn't occupied at present. The status of the chair or the house (or where ever it may be) after a person vacates from it is 'vacant'. So a 'vacant' position means that the position is unoccupied now. It may have been vacated or it could've never been occupied in the first place. A 'vacated' position means that there was somebody who was occupying that position and now, that place is unoccupied. That is the significant difference between the words. This is the Merriam-Webster definition of vacant and vacate.