What is `n(AuuBuuC)?`

Answer

To find \( n(A \cup B \cup C) \), we will use the principle of inclusion-exclusion. The formula for the number of elements in the union of three sets \( A \), \( B \), and \( C \) is given by:

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

### Step-by-Step Solution:

1. **Identify the Number of Elements in Each Set**:
- Let \( n(A) \) be the number of elements in set \( A \).
- Let \( n(B) \) be the number of elements in set \( B \).
- Let \( n(C) \) be the number of elements in set \( C \).

2. **Identify the Number of Elements in Intersections**:
- Let \( n(A \cap B) \) be the number of elements common to both sets \( A \) and \( B \).
- Let \( n(A \cap C) \) be the number of elements common to both sets \( A \) and \( C \).
- Let \( n(B \cap C) \) be the number of elements common to both sets \( B \) and \( C \).
- Let \( n(A \cap B \cap C) \) be the number of elements common to all three sets \( A \), \( B \), and \( C \).

3. **Apply the Inclusion-Exclusion Formula**:
Substitute the values into the inclusion-exclusion formula:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

4. **Calculate the Result**:
- Perform the arithmetic operations based on the values you have for each of the sets and their intersections.

5. **Conclusion**:
- The result from the above calculation will give you the total number of unique elements in the union of sets \( A \), \( B \), and \( C \).