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To solve for \( n(B) \), we need to interpret and use the given information. Let’s go over it step-by-step.

### Given Information:
1. \( n(A) = 17 \)
2. \( n(U) = 42 \) (where \( U \) is the universal set)
3. \( n(C) = 22 \)
4. \( n(B) = 2 \times n(A \cap C) = 4 \times n(A \cap B \cap C) \)
5. \( n(A \cap B) = 8 \)
6. \( n(B \cap C) = 9 \)
7. \( n((A \cup B \cup C)^c) = 5 \)
8. \( n(A \cup B \cup C) = 37 \)

### Step 1: Calculate \( n(A \cap C) \) and \( n(A \cap B \cap C) \) from \( n(B) \)

From the fourth given condition:
\[
n(B) = 2 \times n(A \cap C) = 4 \times n(A \cap B \cap C)
\]

This tells us that \( n(A \cap C) \) is half of \( n(B) \) and \( n(A \cap B \cap C) \) is a quarter of \( n(B) \).

Let’s denote \( n(B) \) by \( x \). Then:
\[
n(A \cap C) = \frac{x}{2}
\]
\[
n(A \cap B \cap C) = \frac{x}{4}
\]

### Step 2: Use the Union Formula for Three Sets

The formula for the union of three sets is:
\[
n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)
\]

Substitute the known values and expressions:
\[
37 = 17 + x + 22 - 8 - \frac{x}{2} - 9 + \frac{x}{4}
\]

### Step 3: Simplify the Equation

Combine like terms:
\[
37 = 31 + x - \frac{x}{2} + \frac{x}{4}
\]

Combine terms involving \( x \) on the right side:
\[
37 = 31 + \left( x - \frac{x}{2} + \frac{x}{4} \right)
\]

Rewrite \( x - \frac{x}{2} + \frac{x}{4} \) as a single fraction:
\[
x - \frac{x}{2} + \frac{x}{4} = \frac{4x - 2x + x}{4} = \frac{3x}{4}
\]

So now we have:
\[
37 = 31 + \frac{3x}{4}
\]

### Step 4: Solve for \( x \)

Subtract 31 from both sides:
\[
6 = \frac{3x}{4}
\]

Multiply both sides by \(\frac{4}{3}\):
\[
x = 8
\]

### Conclusion:
\[
n(B) = 8
\]

---

Would you like further details on any of these steps, or have any questions?

### Related Questions:
1. How would the solution change if \( n(A \cap B) \) were different?
2. What if the value of \( n(U) \) changed? How would it affect the calculation?
3. How does the formula for \( n(A \cup B \cup C) \) apply if only two sets are involved?
4. Can we calculate \( n(A \cap C) \) and \( n(A \cap B \cap C) \) directly with \( n(B) = 8 \)?
5. What would happen if \( n((A \cup B \cup C)^c) \) were larger?

### Tip:
Always substitute values and simplify expressions carefully to avoid algebraic errors, especially when using formulas with multiple terms like the union of sets.

Given: n(A) = 17, n(U) = 42, n(C) = 22, n(B) = 2 * n(A ∩ C) = 4 * n(A ∩ B ∩ C), n(A ∩ B) = 8, n(B ∩ C) = 9, n[(A ∪ B ∪ C)^c] = 5, n(A ∪ B ∪ C) = 37. Find n(B).

Learn how to solve for n(B) in a set theory problem involving union and intersection of sets with given values. This detailed solution covers key concepts in set theory and algebra, suitable for students in Grades 9-12.

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