Let's solve this problem using set theory, specifically with the principle of inclusion-exclusion.

Definitions:

• Let $J$ be the set of students who know Java.
• Let $C++$ be the set of students who know C++.
• Let $C$ be the set of students who know C.

We have the following information:

1. Total number of students, $n(U) = 50$.
2. Students who know Java, $|J| = 30$.
3. Students who know C++, $|C++| = 18$.
4. Students who know C, $|C| = 26$.
5. Students who know both Java and C++, $|J \cap C++| = 9$.
6. Students who know both Java and C, $|J \cap C| = 16$.
7. Students who know both C++ and C, $|C++ \cap C| = 8$.
8. Students who know at least one language, $|J \cup C++ \cup C| = 47$.

Step-by-Step Solution

To find the number of students who know none of these languages, we calculate:

$\text{Number of students who know none} = n(U) - |J \cup C++ \cup C|$

Since $n(U) = 50$ and $|J \cup C++ \cup C| = 47$, we have:

$\text{Number of students who know none} = 50 - 47 = 3$

Final Answer

3 students know none of the three languages.

Would you like details on how the inclusion-exclusion principle was applied here?

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Here are some related questions:

1. How many students know only Java?
2. How many students know all three languages?
3. How many students know exactly two languages?
4. How many students know only C++?
5. How many students know only one language?

Tip: When tackling problems with multiple overlapping sets, the inclusion-exclusion principle helps by subtracting out overlaps to avoid double-counting.