If `n(A) = p and n(B) = q` and no. of subsets of `A` are `48` more than the no. of subsets of `B` then :

Options

A. `p=6,q=5`
B. `p=6,q=4`
C. `p=5,q=6`
D. `p=4,q=6`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We are given two sets, A and B, with the number of elements in set A denoted as \( n(A) = p \) and the number of elements in set B denoted as \( n(B) = q \). We know that the number of subsets of set A is 48 more than the number of subsets of set B.

### Step 2: Write the Formula for Subsets
The number of subsets of a set with \( n \) elements is given by the formula:
\[ \text{Number of subsets} = 2^n \]
Thus, for set A, the number of subsets is:
\[ \text{Number of subsets of A} = 2^p \]
And for set B, the number of subsets is:
\[ \text{Number of subsets of B} = 2^q \]

### Step 3: Set Up the Equation
According to the problem, the number of subsets of A is 48 more than the number of subsets of B. Therefore, we can set up the following equation:
\[ 2^p = 2^q + 48 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us:
\[ 2^p - 2^q = 48 \]

### Step 5: Factor the Left Side
We can factor the left side of the equation:
\[ 2^q (2^{p-q} - 1) = 48 \]

### Step 6: Analyze the Factors
Since \( 48 = 16 \times 3 = 2^4 \times 3 \), we can analyze the possible values for \( 2^q \) and \( 2^{p-q} - 1 \). The possible values for \( 2^q \) can be \( 1, 2, 4, 8, 16 \) (since \( 48 \) is a product of \( 2^4 \)).

### Step 7: Test Possible Values
1. **If \( 2^q = 16 \)**, then:
\[ 2^{p-q} - 1 = 3 \implies 2^{p-q} = 4 \implies p - q = 2 \]
Thus, \( q = 4 \) and \( p = 6 \).

2. **If \( 2^q = 8 \)**, then:
\[ 2^{p-q} - 1 = 6 \implies 2^{p-q} = 7 \quad (\text{not a power of 2}) \]

3. **If \( 2^q = 4 \)**, then:
\[ 2^{p-q} - 1 = 12 \implies 2^{p-q} = 13 \quad (\text{not a power of 2}) \]

4. **If \( 2^q = 2 \)**, then:
\[ 2^{p-q} - 1 = 24 \implies 2^{p-q} = 25 \quad (\text{not a power of 2}) \]

5. **If \( 2^q = 1 \)**, then:
\[ 2^{p-q} - 1 = 48 \implies 2^{p-q} = 49 \quad (\text{not a power of 2}) \]

### Step 8: Conclusion
From the analysis, the only valid solution is \( p = 6 \) and \( q = 4 \). Therefore, the correct answer is \( p = 6 \) and \( q = 4 \).