Let's solve this step-by-step using the Inclusion-Exclusion Principle.

We are given:

• Total number of investors: $700$
• Investors who had not invested in either stocks or bonds: $87$
• Investors who invested in stocks: $n(S) = 440$
• Investors who invested in bonds: $n(B) = 384$
• Investors who invested in either stocks or bonds: $n(S \cup B) = 700 - 87 = 613$

Using the Inclusion-Exclusion Principle, we have the formula: $n(S \cup B) = n(S) + n(B) - n(S \cap B)$ Substituting the values into this equation: $613 = 440 + 384 - n(S \cap B)$ Now, solve for $n(S \cap B)$: $613 = 824 - n(S \cap B)$ $n(S \cap B) = 824 - 613$ $n(S \cap B) = 211$

Conclusion:

The number of investors who had invested in both stocks and bonds is 211.

Would you like a detailed explanation on any part of this solution? Here are some related questions you might find useful:

1. How does the Inclusion-Exclusion Principle work in set theory?
2. What if some investors had invested in neither, but also in other categories like real estate? How would that affect the calculation?
3. How can we visualize this problem using a Venn diagram?
4. Can the formula be extended for three or more sets, such as adding a third investment type?
5. What if we only knew the number of investors in both categories but not individual counts for stocks or bonds?

Tip: The Inclusion-Exclusion Principle is a versatile tool in counting problems, especially when dealing with overlapping sets.