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This question is asking to determine the total number of students in a class by analyzing a Venn diagram, which represents the number of students who fall into one or more of three categories: International, Scholarship-awardees, or Industry-sponsored. Here's the breakdown from the image:

1. **Categories and overlaps (as per the Venn diagram):**
   - International students = 100 total
   - Scholarship-awardees = 200 total
   - Industry-sponsored = 210 total
   - The overlaps shown are:
     - International and Scholarship = 40
     - Scholarship and Industry-sponsored = 30
     - International and Industry-sponsored = 25
     - All three categories = 5 students

### Let's calculate the total number of students using the principle of inclusion and exclusion:

\[
\text{Total students} = (\text{International}) + (\text{Scholarship}) + (\text{Industry-sponsored}) - (\text{overlap of two categories}) + (\text{overlap of all three categories})
\]

Where:
- The individual categories include students in only those categories, so we subtract overlaps and then add back the three-way overlap to avoid double-counting.

### Calculation:
- Students in International = 100
- Students in Scholarship = 200
- Students in Industry-sponsored = 210
- Subtract the overlap of two categories:
  - International ∩ Scholarship = 40
  - Scholarship ∩ Industry-sponsored = 30
  - International ∩ Industry-sponsored = 25
- Add the overlap of all three categories (counted 3 times before) = 5

The formula becomes:

\[
\text{Total} = 100 + 200 + 210 - 40 - 30 - 25 + 5
\]
\[
\text{Total} = 485 \text{ students}
\]

So, the total number of students in the class is **485**.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What is the principle of inclusion-exclusion and how is it used in set theory?
2. How would the total number of students change if the overlap of all three categories increased?
3. Can you calculate the number of students that belong only to one category in the class?
4. How would you modify this formula if there were more categories or different overlaps?
5. What role does double-counting play in these types of problems?

**Tip:** When using Venn diagrams, it's essential to track how many times each intersection is counted to avoid double-counting, especially when overlapping multiple categories.

In a class, all students fall in one or more of the following categories: International, Scholarship-awardees, or Industry-sponsored. How many total students are there in the class based on the Venn diagram and the given information?

Calculate the total number of students in a class using a Venn diagram and the Inclusion-Exclusion Principle. Learn how to analyze overlaps between categories like International, Scholarship-awardees, and Industry-sponsored students. The problem covers key mathematical concepts such as set theory and the inclusion-exclusion principle. Suitable for high school students in Grades 10-12.

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