Answered: The Enormous State University History…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.6: Permutations And Combinations

Problem 1SC: If a man has 4 sweaters and 5 pairs of slacks, how many different outfit can he wear?

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Question

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 44 sections total, there are 4,600 students who would like to take a course, and there are 57 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?

Ancient History sections

Medieval History sections

Modern History sections

Expert Solution

Step 1

Let x represents Ancient History sections, y represents Medieval History sections and z represents Modern History sections.

The department is allowed to offer 44 sections total.

That is, $x + y + z = 44$ .

Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section and the total professors are 57.

$x + y + 2 z = 57 .$

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 and total students are 4600.

That is, $100 x + 50 y + 200 z = 4600$ .

Therefore, the system of equations is,

$\begin{array}{rcl} x + y + z & = & 44 \dots (1) \\ x + y + 2 z & = & 57 \dots (2) \\ 100 x + 50 y + 200 z & = & 4600 \dots (3) \end{array}$