It is stated in the problem that
n(S) = 440
and
n(B) = 384.

To find the number of investors who had invested in both stocks and bonds, use the Inclusion-Exclusion Principle, given as
n(S ∪ B) = n(S) + n(B) − n(S ∩ B).

Substitute the given information and
n(S ∪ B) = 613
into this formula.
613 = 440 +
− n(S ∩ B)

To solve this problem using the **Inclusion-Exclusion Principle**, we're given:

- \( n(S) = 440 \) (the number of investors in stocks),
- \( n(B) = 384 \) (the number of investors in bonds),
- \( n(S \cup B) = 613 \) (the total number of investors who invested in either stocks or bonds or both).

We need to find \( n(S \cap B) \), the number of investors who invested in both stocks and bonds.

### Inclusion-Exclusion Principle formula:
\[
n(S \cup B) = n(S) + n(B) - n(S \cap B)
\]
Substitute the given values into this formula:
\[
613 = 440 + 384 - n(S \cap B)
\]

Now, simplify:
\[
613 = 824 - n(S \cap B)
\]

Solving for \( n(S \cap B) \):
\[
n(S \cap B) = 824 - 613
\]
\[
n(S \cap B) = 211
\]

Thus, the number of investors who invested in both stocks and bonds is **211**.

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Do you want more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the Inclusion-Exclusion Principle used for in set theory?
2. How would the formula change if we had three sets, like stocks, bonds, and real estate?
3. What would happen if there were no investors who invested in both stocks and bonds?
4. How would you represent the complement of \( n(S \cap B) \) in this scenario?
5. Can this principle be applied to probabilities instead of counting?

**Tip**: When solving problems involving sets, always break down the problem using Venn diagrams if you struggle to visualize the overlap between groups.

It is stated in the problem that n(S) = 440 and n(B) = 384. To find the number of investors who had invested in both stocks and bonds, use the Inclusion-Exclusion Principle, given as n(S ∪ B) = n(S) + n(B) − n(S ∩ B). Substitute the given information and n(S ∪ B) = 613 into this formula. 613 = 440 + − n(S ∩ B).

Solve a set theory problem using the Inclusion-Exclusion Principle to find the number of investors who invested in both stocks and bonds. With n(S) = 440, n(B) = 384, and n(S ∪ B) = 613, discover how to calculate the overlap in this detailed, step-by-step guide. Suitable for Grades 10-12.

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