If n(A)=25+x, n(B) = 27 - x, and `n(A uuB)=46`, then n `(A nnB)=`

Answer

To solve the problem, we will use the formula that relates the number of elements in the union and intersection of two sets. The formula is:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Given:
- \( n(A) = 25 + x \)
- \( n(B) = 27 - x \)
- \( n(A \cup B) = 46 \)

We need to find \( n(A \cap B) \).

### Step-by-step Solution:

1. **Write the formula for the union of sets**:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

2. **Substitute the known values into the formula**:
\[ 46 = (25 + x) + (27 - x) - n(A \cap B) \]

3. **Simplify the equation**:
- Combine like terms:
\[ 46 = 25 + 27 + x - x - n(A \cap B) \]
- The \( x \) and \( -x \) cancel each other out:
\[ 46 = 52 - n(A \cap B) \]

4. **Rearrange the equation to isolate \( n(A \cap B) \)**:
\[ n(A \cap B) = 52 - 46 \]

5. **Calculate \( n(A \cap B) \)**:
\[ n(A \cap B) = 6 \]

Thus, the number of elements in the intersection of sets A and B is \( n(A \cap B) = 6 \).

### Final Answer:
\[ n(A \cap B) = 6 \]