Consider a set P consisting of 5 elements . A sub set .A. of P is chosen thereafter set .P. is reconstructed and finally another sub set .B. is chosen from P. The number of ways of choosing .A. and .B. such that `(A cup B) ne P` is :

Explanation

To solve the problem of finding the number of ways to choose subsets \( A \) and \( B \) from a set \( P \) with 5 elements such that \( A \cup B \neq P \), we can follow these steps:

### Step 1: Calculate the total number of ways to choose subsets \( A \) and \( B \)

Each element in the set \( P \) can belong to:
- Subset \( A \)
- Subset \( B \)
- Both subsets \( A \) and \( B \)
- Neither subset

This gives us 4 choices for each of the 5 elements in set \( P \). Therefore, the total number of ways \( m \) to choose subsets \( A \) and \( B \) is:

\[ m = 4^5 \]

### Step 2: Calculate the number of ways where \( A \cup B = P \)

For the condition \( A \cup B = P \) to hold, each element must belong to at least one of the subsets \( A \) or \( B \). Thus, for each of the 5 elements, we have 3 choices:
- The element belongs to subset \( A \) only
- The element belongs to subset \( B \) only
- The element belongs to both subsets \( A \) and \( B \)

Therefore, the total number of ways \( n \) to choose subsets \( A \) and \( B \) such that \( A \cup B = P \) is:

\[ n = 3^5 \]

### Step 3: Calculate the number of ways where \( A \cup B \neq P \)

To find the number of ways where \( A \cup B \neq P \), we subtract the number of ways where \( A \cup B = P \) from the total number of ways:

\[ \text{Number of ways where } A \cup B \neq P = m - n = 4^5 - 3^5 \]

### Step 4: Calculate the values of \( 4^5 \) and \( 3^5 \)

Calculating \( 4^5 \):

\[ 4^5 = 1024 \]

Calculating \( 3^5 \):

\[ 3^5 = 243 \]

### Step 5: Perform the subtraction

Now, we can find the final result:

\[ 1024 - 243 = 781 \]

### Final Answer

The number of ways to choose subsets \( A \) and \( B \) such that \( A \cup B \neq P \) is:

\[ \boxed{781} \]

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