If A and B are two sets such that `n(A) = 17,n(B)=23 and n(A cup B) = 38`, then find `n(A cap B)`.

Answer

To find the number of elements in the intersection of sets A and B, we can use the formula for the union of two sets. The formula is:

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

Where:
- \( n(A \cup B) \) is the number of elements in the union of sets A and B.
- \( n(A) \) is the number of elements in set A.
- \( n(B) \) is the number of elements in set B.
- \( n(A \cap B) \) is the number of elements in the intersection of sets A and B.

### Step-by-step Solution:

1. **Identify the given values:**
- \( n(A) = 17 \)
- \( n(B) = 23 \)
- \( n(A \cup B) = 38 \)

2. **Substitute the values into the formula:**
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Plugging in the known values:
\[ 38 = 17 + 23 - n(A \cap B) \]

3. **Calculate the sum of \( n(A) \) and \( n(B) \):**
\[ 17 + 23 = 40 \]
So, we have:
\[ 38 = 40 - n(A \cap B) \]

4. **Rearrange the equation to solve for \( n(A \cap B) \):**
\[ n(A \cap B) = 40 - 38 \]

5. **Perform the subtraction:**
\[ n(A \cap B) = 2 \]

### Final Answer:
The number of elements in the intersection of sets A and B is \( n(A \cap B) = 2 \).

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