If n(A) = 10 ,n(B) = 20 and `n(A cup B) = 15`, then find `n(A cap B)` .

Options

A. 5
B. 10
C. 15
D. 20

Answer

To find the number of elements in the intersection of sets A and B, denoted as \( n(A \cap B) \), we can use the formula for the union of two sets:

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

### Step-by-Step Solution:

1. **Identify the given values**:
- \( n(A) = 10 \)
- \( n(B) = 20 \)
- \( n(A \cup B) = 15 \)

2. **Substitute the known values into the formula**:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Plugging in the values we have:
\[ 15 = 10 + 20 - n(A \cap B) \]

3. **Simplify the equation**:
\[ 15 = 30 - n(A \cap B) \]

4. **Rearrange the equation to solve for \( n(A \cap B) \)**:
\[ n(A \cap B) = 30 - 15 \]

5. **Calculate the value**:
\[ n(A \cap B) = 15 \]

### Final Answer:
Thus, the number of elements in the intersection of sets A and B is \( n(A \cap B) = 15 \).

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