There are 600 elements in universal set U. If A and B are subsets of U and `n(A) = 300, n(B) = 250 and n(A cap B) = 50`, then `n(A' cap B')` is :

Options

A. 50
B. 100
C. 150
D. 200

Answer

To solve the problem step by step, we will utilize the information provided about the universal set and the subsets A and B. We will also apply De Morgan's laws to find the required intersection of the complements.

### Step-by-Step Solution:

1. **Identify the given values:**
- Number of elements in the universal set \( U \): \( n(U) = 600 \)
- Number of elements in set \( A \): \( n(A) = 300 \)
- Number of elements in set \( B \): \( n(B) = 250 \)
- Number of elements in the intersection of sets \( A \) and \( B \): \( n(A \cap B) = 50 \)

2. **Apply De Morgan's Law:**
- According to De Morgan's Law, we have:
\[ n(A' \cap B') = n((A \cup B)') \]
- This means that to find \( n(A' \cap B') \), we first need to find \( n(A \cup B) \).

3. **Calculate \( n(A \cup B) \):**
- We can use the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
- Substituting the known values:
\[ n(A \cup B) = 300 + 250 - 50 \]
- Performing the calculation:
\[ n(A \cup B) = 500 \]

4. **Calculate \( n(A' \cap B') \):**
- Now we can find \( n((A \cup B)') \):
\[ n((A \cup B)') = n(U) - n(A \cup B) \]
- Substituting the values:
\[ n((A \cup B)') = 600 - 500 \]
- Performing the calculation:
\[ n((A \cup B)') = 100 \]

5. **Conclusion:**
- Therefore, the number of elements in \( n(A' \cap B') \) is:
\[ n(A' \cap B') = 100 \]

### Final Answer:
\[ n(A' \cap B') = 100 \]