U is a universal set and n `(U) = 160` where `A ,B` and `C ` are subset of `U`. If ` n(A) = 50, n (B) = 70 , n(BuuC) = phi , n (BnnC) = 15` and `AuuBuu C = U`, then `n (C )` equals,

Options

A. 40
B. 50
C. 55
D. 60

Explanation

To solve the problem step by step, we will use the information provided and the principles of set theory.

### Step 1: Understand the Given Information
We have:
- Universal set \( U \) with \( n(U) = 160 \)
- Subsets \( A, B, C \) with:
- \( n(A) = 50 \)
- \( n(B) = 70 \)
- \( n(B \cup C) = \phi \) (which means \( B \) and \( C \) have no elements in common)
- \( n(B \cap C) = 15 \)
- \( A \cup B \cup C = U \)

### Step 2: Use the Formula for Union of Sets
We know the formula for the union of two sets:
\[ n(B \cup C) = n(B) + n(C) - n(B \cap C) \]

### Step 3: Substitute the Known Values
From the problem, we substitute the known values into the formula:
\[ n(B \cup C) = n(B) + n(C) - n(B \cap C) \]
Substituting the values:
\[ \phi = 70 + n(C) - 15 \]

### Step 4: Simplify the Equation
Now we simplify the equation:
\[ \phi = 70 + n(C) - 15 \]
\[ \phi = 55 + n(C) \]

### Step 5: Solve for \( n(C) \)
Since \( n(B \cup C) = \phi \), we can set the equation to zero:
\[ 0 = 55 + n(C) \]
Now, solving for \( n(C) \):
\[ n(C) = -55 \]

### Step 6: Analyze the Result
The result \( n(C) = -55 \) is not possible because the number of elements in a set cannot be negative. This indicates that there is an inconsistency in the provided data.

### Conclusion
Given the conditions and the calculations, we conclude that there is no possible value for \( n(C) \) that satisfies all the conditions.