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.
not a (used before a noun to convey the opposite of the noun's meaning): She's no beginner on the ski slopes. noun, plural noes, nos. 6. an utterance of the word âno.â. 7. a denial or refusal: He responded with a definite no.