$(20$ points$)$ Consider a final exam scheduling problem where the binary decision variables have been defined as $x_{t,}=1$ if course $i\text{'}$s exam is scheduled in time period t; $0$ otherwise. Let's assume that there are $30$ exam periods where the first $15$ are in the first week, and the rest of the periods are in the second week. The total number of courses is C$.$ Formulate the following constraints using only the above defined notation:

a$.$ The exam of course $i=540$ cannot be scheduled during the first week

b$.$ In each exam period, there should be at most $70$ exams.

c$.$ The total number of exams during the second week should be at most $500.$

d$.$ The total number of exams in the first week should be at least as many as the total number of exams in the second week.