Let `n(U)=700`, `n(A)=200`,`n(B)=300` and `n(AnnB)=100`, then find `n(A'nnB')`

Answer

To solve the problem, we need to find \( n(A' \cap B') \), which represents the number of elements in the complement of set A and the complement of set B. We can use the principle of inclusion-exclusion and the properties of complements.

### Step-by-Step Solution:

1. **Identify Given Values**:
- \( n(U) = 700 \) (Total number of elements in the universal set)
- \( n(A) = 200 \) (Number of elements in set A)
- \( n(B) = 300 \) (Number of elements in set B)
- \( n(A \cap B) = 100 \) (Number of elements in the intersection of sets A and B)

2. **Calculate \( n(A \cup B) \)**:
Using the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substitute the known values:
\[ n(A \cup B) = 200 + 300 - 100 = 400 \]

3. **Calculate \( n(A' \cap B') \)**:
The complement of the union of A and B can be found using the universal set:
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
Substitute the values we have:
\[ n(A' \cap B') = 700 - 400 = 300 \]

### Final Answer:
\[ n(A' \cap B') = 300 \]