Question

In a student survey, 200 indicated that they would attend Summer Session I, and 150 indicated Summer Session II. If 75 students plan to attend both summer sessions, and 275 indicated that they would attend neither session, how many students participated in the survey?

Short answer

550 students participated in the survey.

Step-by-step solution

Define Variables

Let A be the number of students who will attend Summer Session I, B be the number of students who will attend Summer Session II, and C be the number of students who will attend both sessions.

Write Down the Given Numbers

Given: - A = 200 (students for Summer Session I)- B = 150 (students for Summer Session II)- C = 75 (students for both sessions)- Students attending neither session = 275

Calculate Students Attending At Least One Session

Use the principle of inclusion and exclusion to find the number of students attending at least one session. Formula: \[ |A \cup B| = |A| + |B| - |A \cap B| \]Substitute the values: \[ |A \cup B| = 200 + 150 - 75 = 275 \]

Calculate Total Number of Students in the Survey

Add the number of students attending at least one session to the number of students attending neither session: \[ \text{Total Students} = |A \cup B| + \text{Neither} = 275 + 275 = 550 \]

Key concepts

Principle of Inclusion and Exclusion

The principle of inclusion and exclusion is a fundamental concept in combinatorics. It's useful for counting the number of elements in the union of multiple sets. This principle is used to correct over-counting when sets overlap. Here’s the basic idea:
When we want to find the number of elements in either set A or set B, we add the number of elements in A to the number of elements in B. But wait! If we do this directly, we count the elements in the intersection of both sets (those in both A and B) twice. So, we need to subtract the number of elements in the intersection.
The formula looks like this:
\(|A \cup B| = |A| + |B| - |A \cap B|\)
In the context of our student survey problem, the formula helps determine the number of students attending at least one summer session:
\(|A \cup B| = 200 + 150 - 75 = 275\)
This result tells us that 275 students attend at least one session.

Set Theory in Mathematics

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It provides a foundation for various other fields of mathematics.
In our exercise, we use sets to categorize students based on the sessions they plan to attend. Here’s a quick rundown of important set theory concepts:

• Union (\[A \cup B\]): The set of elements in either set A, set B, or both.
• Intersection (\[A \cap B\]): The set of elements in both set A and set B.
• Complement: The set of elements not in a particular set.

For our problem, set A represents students attending Summer Session I and set B represents students attending Summer Session II. The principle of inclusion and exclusion helps us efficiently manage these groups and tackle overlaps.
Considering the given data:

1. \[ A = 200 \] (students for Summer Session I)
2. \[ B = 150 \] (students for Summer Session II)
3. \[ A \cap B = 75 \] (students for both sessions)
We quickly find that the union (students attending at least one session) is: \[ |A \cup B| = 275 \].

Solving Survey Problems

Survey problems often require us to manage data efficiently and understand the relationships between different groups of participants. Here’s how we solve them using the principles discussed:
1. **Define the groups (sets)**: Identify the different groups of participants based on the survey’s criteria (A: Summer Session I, B: Summer Session II).
2. **Use the principles and formulas**: Apply the principle of inclusion and exclusion to manage overlaps and avoid double-counting. In our problem, we use the union formula to find the total attending at least one session.
3. **Sum the groups properly**: After calculating how many are in at least one group, don't forget those in neither group. Add them together for the total surveyed.
Example walkthrough:
- Given:
\[ \begin{align*} & A = 200 \text{ (Session I attendees)}\ \ \ \ \ \ ewline \ & B = 150 \text{ (Session II attendees)}\ \ \ \ \ \ ewline \ & A \cap B = 75 \text{ (Both sessions)}\ \ \ \ \ \ ewline \ & Neither session = 275\end{align*} \]
- Find attending at least one session using inclusion and exclusion:\
\[|A \cup B| = 200 + 150 - 75 = 275\]

- Calculate the total by adding those attending neither session:
\[ \begin{align*} & \text{Total Students} = |A \cup B| + \text{Neither} = 275 + 275 = 550\ \ \ \ \ \ ewline \end{align*} \]
Thus, 550 students participated in the survey. By consistently applying set theory and the principle of inclusion and exclusion, solving survey problems becomes a structured process.