The Enormous State University History Department offers three courses$$Ancient$,$ Medieval, and Modern History$$and the chairperson is trying to decide how many sections of each to offer this semester. The department is allowed to offer $47$ sections total, there are $5,000$ students who would like to take a course, and there are $61$ professors to teach them. Sections of Ancient History have $100$ students each, sections of Medieval History hold $50$ students each, and sections of Modern History have $200$ students each. Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section. How many sections of each course should the chair schedule in order to offer all the sections that they are allowed, accommodate all of the students, and give one teaching assignment to each professor? HINT $[$See Example $2.]($Assume each section has the maximum number of students allowed.$)$