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.
NM, meaning No Message. Also written as N/M, n/m, or *n/m*. Used when the entire content of the email is contained in the subject and the body remains empty. This saves the recipient's time because she then does not have to open the email.