Question

The enrollment for the four courses Biol212, Poli115, Econ313, and Fina215 is 108 . $203,315,$ and $212,$ respectively. No student is in all four of these courses. No student is in the three courses Biology 212 , Fina 215 , and Poli 115 . No student takes $\mathrm{E}\mathrm{con}313$ and Fina 215 in the same semester. Polit 15 and Fina 215 are not allowed in the same term. There are 39 students in both Biol212 and Poli115, and 48 students in both Polit 15 and Econ313 as well as in the two courses Biol2 12 and Econ313. Biol212, Polit 15 . and $\mathrm{F}\mathrm{con}313$ have a common enrollment of $73.$ Biol 212 and Fina 215 have a common enrollment of $67.$ How many different students are enrolled in these four courses?

Short answer

There are 636 different students enrolled in these courses.

Step-by-step solution

Understand the Problem

We have four courses with varying enrollments and overlapping students in multiple courses. We'll use the inclusion-exclusion principle to find the total number of unique students.

Identify Known Values

List the total enrollments: Biol212 = 108, Poli115 = 203, Econ313 = 315, Fina215 = 212. List overlaps: Biol212 & Poli115 = 39, Poli115 & Econ313 = 48, Biol212 & Econ313 = 48, Biol212, Poli115 & Econ313 = 73, and Biol212 & Fina215 = 67.

List Additional Constraints

Know that no student is in all four courses, none are in Biol212, Fina215, & Poli115, no student takes Econ313 & Fina215 together, and Poli115 & Fina215 can't be in the same term.

Use Inclusion-Exclusion Principle

Using inclusion-exclusion, calculate the number of unique students as: $|A\cup B\cup C\cup D|=|A|+|B|+|C|+|D|-|AB|-|AC|-|AD|-|BC|-|BD|-|CD|+|ABC|$.

Fill in Values and Simplify

Substitute the given values: $$108+203+315+212-39-48-67-48-73-0$$ and add $73$ again for Biol212, Poli115, and Econ313.

Calculate Unique Students

Calculate: $838-275+73=636$ students.

Key concepts

Overlapping Students

Understanding overlapping students is crucial in determining how many unique individuals are part of several groups or sets. In this context, sets refer to different courses, and overlaps occur when a student is enrolled in more than one course.

To identify overlapping students, we first need to figure out which groups have shared members. For example, in the given exercise, multiply-enrolled students appear in Biol212 & Poli115, Poli115 & Econ313, and Biol212 & Econ313, among others. This creates subsets of students who belong to two or more courses simultaneously.

In real-world applications, considering overlapping students helps:

• Ensure accurate resource allocation, like classroom size and teaching assistants.
• Plan schedules to minimize conflicts for students taking multiple courses.
• Improve course performance by understanding student commitments across courses.

Course Enrollment

Course enrollment refers to the process of students signing up for educational courses offered by an institution. In the exercise, four courses have varying numbers of enrollments: Biol212 with 108 students, Poli115 with 203 students, Econ313 with 315 students, and Fina215 with 212 students. However, the total number of unique students is less due to overlaps.

When managing course enrollments, institutions must consider several factors:

• Capacity limitations: Ensuring that courses do not exceed the room or instructor limitations.
• Prerequisite checks: Making sure students meet the required prerequisites before enrolling.
• Conflict resolution: Adjusting overlaps in schedules to accommodate students enrolled in multiple courses.

Effective course enrollment management can enhance the educational experience, offering both students and instructors a more organized and efficient learning environment.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with discrete elements that use algebra and arithmetic. It includes the study of mathematical structures that are fundamentally discrete, such as integers, graphs, and statements in logic.

In this exercise, the Inclusion-Exclusion Principle from Discrete Mathematics is used to calculate the total number of unique students enrolled in multiple courses. This principle helps in finding the number of elements in the union of several sets by considering both individual and overlapping memberships.

The formula used is:$$|A\cup B\cup C\cup D|=|A|+|B|+|C|+|D|-|AB|-|AC|-|AD|-|BC|-|BD|-|CD|+|ABC|$$
This formula sums up all the individual sets, subtracts the pairwise intersections to avoid double-counting, and then adds back the intersections of three sets. Discrete Mathematics thus provides powerful tools for solving complex real-world problems involving discrete elements and relationships.