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.
Definition of no. 1a chiefly Scotland : notb âused as a function word to express the negative of an alternative choice or possibility shall we go out to dinner or no. 2 : in no respect or degree âused in comparisons you're no better than the rest of us.