Two finite sets have m and n elements respectively. The total number of subsets of the first set is 192 more than the total number of subsets of the second set. The values of m and n respectively are (i) 7,6 (ii) 8,6 (iii) 8,5 (iv) 9,7

Options

A. 7,6
B. 8,6
C. 8,5
D. 9,7

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) based on the information given about the subsets of two finite sets.

### Step 1: Understand the relationship between subsets and elements
The number of subsets of a set with \( m \) elements is given by \( 2^m \), and the number of subsets of a set with \( n \) elements is given by \( 2^n \). According to the problem, the total number of subsets of the first set is 192 more than that of the second set.

### Step 2: Set up the equation
From the information given, we can write the equation:
\[ 2^m = 2^n + 192 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ 2^m - 2^n = 192 \]

### Step 4: Factor the left-hand side
We can factor the left-hand side:
\[ 2^n(2^{m-n} - 1) = 192 \]

### Step 5: Express 192 as a product of powers of 2
Next, we can express 192 in terms of powers of 2:
\[ 192 = 64 \times 3 = 2^6 \times 3 \]

### Step 6: Set up the equation with powers of 2
From the factored equation, we can equate:
\[ 2^n(2^{m-n} - 1) = 2^6 \times 3 \]
This implies that \( 2^n = 64 = 2^6 \) and \( 2^{m-n} - 1 = 3 \).

### Step 7: Solve for \( n \)
From \( 2^n = 2^6 \), we find:
\[ n = 6 \]

### Step 8: Solve for \( m \)
Now, from \( 2^{m-n} - 1 = 3 \):
\[ 2^{m-n} = 4 \implies m-n = 2 \]
Substituting \( n = 6 \) into the equation gives:
\[ m - 6 = 2 \implies m = 8 \]

### Final Values
Thus, the values of \( m \) and \( n \) are:
\[ m = 8, \quad n = 6 \]

### Conclusion
The correct option is (ii) 8, 6.