Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The value of m and n is

Options

A. `6,3`
B. `7,6`
C. `7,3`
D. none of these

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) given that the total number of subsets of the first set (with \( m \) elements) is 56 more than the total number of subsets of the second set (with \( n \) elements).

### Step-by-Step Solution:

1. **Understanding the Number of Subsets**:
- For a set with \( m \) elements, the number of subsets is given by \( 2^m \).
- For a set with \( n \) elements, the number of subsets is given by \( 2^n \).

2. **Setting Up the Equation**:
- According to the problem, the number of subsets of the first set is 56 more than the number of subsets of the second set. This can be expressed mathematically as:
\[ 2^m = 2^n + 56 \]

3. **Rearranging the Equation**:
- We can rearrange the equation to isolate the terms involving powers of 2:
\[ 2^m - 2^n = 56 \]

4. **Factoring the Left Side**:
- We can factor the left side of the equation:
\[ 2^n (2^{m-n} - 1) = 56 \]

5. **Finding Possible Values**:
- Now, we need to find integer values for \( n \) such that \( 2^n \) divides 56. The divisors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The powers of 2 among these are:
- \( 2^0 = 1 \)
- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)

6. **Testing Values of \( n \)**:
- **If \( n = 3 \)**:
\[ 2^3 (2^{m-3} - 1) = 56 \implies 8(2^{m-3} - 1) = 56 \implies 2^{m-3} - 1 = 7 \implies 2^{m-3} = 8 \implies m - 3 = 3 \implies m = 6 \]
- **If \( n = 2 \)** or **\( n = 1 \)**, we will find that \( m \) will not yield integer values.

7. **Conclusion**:
- The only valid solution is \( m = 6 \) and \( n = 3 \).

### Final Answer:
The values of \( m \) and \( n \) are:
\[ m = 6, \quad n = 3 \]