A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen, the number of ways of choosing so that `(P cup Q)` is a proper subset of A, is

Options

A. `3^(n)`
B. `4^(n)`
C. `4^(n)-2^(n)`
D. `4^(n)-3^(n)`

Explanation

To find the number of ways to choose subsets \( P \) and \( Q \) from a set \( A \) containing \( n \) elements such that \( P \cup Q \) is a proper subset of \( A \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Proper Subset**:
A proper subset of a set \( A \) is a subset that is not equal to \( A \). This means that at least one element of \( A \) must not be included in the subset.

2. **Identifying Choices for Each Element**:
For each element \( A_i \) in the set \( A \), there are four choices when forming subsets \( P \) and \( Q \):
- The element is included in both \( P \) and \( Q \).
- The element is included in \( Q \) but not in \( P \).
- The element is included in \( P \) but not in \( Q \).
- The element is included in neither \( P \) nor \( Q \).

Thus, for each element \( A_i \), there are 4 options.

3. **Calculating Total Combinations**:
Since there are \( n \) elements in set \( A \), the total number of combinations of subsets \( P \) and \( Q \) is:
\[ 4^n \]

4. **Excluding Non-Proper Subsets**:
We need to exclude the cases where \( P \cup Q \) is not a proper subset of \( A \). This occurs when at least one element of \( A \) is included in \( P \cup Q \). If we consider an element \( A_i \) that belongs to \( P \cup Q \), it can have only three choices:
- The element is included in both \( P \) and \( Q \).
- The element is included in \( Q \) but not in \( P \).
- The element is included in \( P \) but not in \( Q \).

Therefore, if an element is in \( P \cup Q \), it has 3 options.

5. **Calculating Non-Proper Combinations**:
For \( n \) elements, the number of ways to choose subsets such that at least one element is included in \( P \cup Q \) is:
\[ 3^n \]

6. **Finding Proper Subset Combinations**:
To find the number of ways to choose \( P \) and \( Q \) such that \( P \cup Q \) is a proper subset of \( A \), we subtract the non-proper combinations from the total combinations:
\[ \text{Number of ways} = 4^n - 3^n \]

### Conclusion:
The number of ways to choose subsets \( P \) and \( Q \) such that \( P \cup Q \) is a proper subset of \( A \) is:
\[ 4^n - 3^n \]