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. 7, 6
B. 6, 3
C. 5, 1
D. 8, 7

Explanation

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 is 56 more than the total number of subsets of the second set.

### Step-by-step Solution:

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

2. **Setting Up the Equation**:
According to the problem, the total number of subsets of the first set is 56 more than the total 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**:
Since \( 56 \) can be factored into \( 2^3 \times 7 \), we can set \( 2^n \) to be one of the factors of \( 56 \). The possible values for \( 2^n \) are \( 1, 2, 4, 8 \) (which correspond to \( n = 0, 1, 2, 3 \)).

6. **Testing Values of \( n \)**:
- **If \( n = 3 \)**:
\[ 2^n = 8 \Rightarrow 8(2^{m-3} - 1) = 56 \Rightarrow 2^{m-3} - 1 = 7 \Rightarrow 2^{m-3} = 8 \Rightarrow m - 3 = 3 \Rightarrow m = 6 \]
- **If \( n = 2 \)**:
\[ 2^n = 4 \Rightarrow 4(2^{m-2} - 1) = 56 \Rightarrow 2^{m-2} - 1 = 14 \Rightarrow 2^{m-2} = 15 \text{ (not a power of 2)} \]
- **If \( n = 1 \)**:
\[ 2^n = 2 \Rightarrow 2(2^{m-1} - 1) = 56 \Rightarrow 2^{m-1} - 1 = 28 \Rightarrow 2^{m-1} = 29 \text{ (not a power of 2)} \]
- **If \( n = 0 \)**:
\[ 2^n = 1 \Rightarrow 1(2^{m-0} - 1) = 56 \Rightarrow 2^m - 1 = 56 \Rightarrow 2^m = 57 \text{ (not a power of 2)} \]

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

Thus, the values of \( m \) and \( n \) are:
\[ \boxed{m = 6, n = 3} \]