A company, Nice Inc., will award $45$ fellowships to high$-$achieving UCR students from four different majors: computer science, biology, political science and history. They decided to give fellowship awards to at least $10$ students majoring in computer science and at most $7$ biology majors. The number of political science and history majors should be between $5$ and $8$ students each. How many possible lists of awardees are there? You need to give a complete derivation for the final answer, using the method developed in class. $($Brute force listing of all lists will not be accepted.$)$