# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-10/16

Exercises: Try Out - Set 10: Real exam simulation to sharpen your skills and build confidence. - Venn Diagram

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## Exercise 16

### Question

export const metadata = {
  title: "Venn Diagram",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

In a class of <InlineMath math="50" /> students, each student has at least one favorite subject among three subjects: Mathematics, Biology, and Physics.

The data is as follows:
- <InlineMath math="35" /> students like Mathematics
- <InlineMath math="20" /> students like Biology
- <InlineMath math="15" /> students like Physics
- <InlineMath math="5" /> students like Physics and Biology
- <InlineMath math="13" /> students like Mathematics and Biology
- <InlineMath math="7" /> students like Mathematics and Physics

How many students like Mathematics, Biology, and Physics?


### Choices

- [ ] $$1$$
- [ ] $$2$$
- [ ] $$3$$
- [ ] $$4$$
- [x] $$5$$

### Answer & Explanation

export const metadata = {
  title: "Venn Diagram",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let <InlineMath math="x" /> be the number of students who like all three subjects (Mathematics, Physics, and Biology).

We can use the inclusion-exclusion principle for sets:

<MathContainer>
  <BlockMath math="\begin{aligned} n(M \cup B \cup P) &= n(M) + n(B) + n(P) \\ &- (n(M \cap B) + n(M \cap P) + n(B \cap P)) \\ &+ n(M \cap B \cap P) \end{aligned}" />
</MathContainer>

Given:

<MathContainer>
  <BlockMath math="n(M \cup B \cup P) = 50" />
  <BlockMath math="n(M) = 35" />
  <BlockMath math="n(B) = 20" />
  <BlockMath math="n(P) = 15" />
  <BlockMath math="n(M \cap B) = 13" />
  <BlockMath math="n(M \cap P) = 7" />
  <BlockMath math="n(B \cap P) = 5" />
  <BlockMath math="n(M \cap B \cap P) = x" />
</MathContainer>

Substitute these values into the formula:

<MathContainer>
  <BlockMath math="50 = 35 + 20 + 15 - (13 + 7 + 5) + x" />
  <BlockMath math="50 = 70 - 25 + x" />
  <BlockMath math="50 = 45 + x" />
  <BlockMath math="x = 5" />
</MathContainer>

Therefore, the number of students who like all three subjects is <InlineMath math="5" />.


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