Two finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. The value of m - n is

Explanation

To solve the problem, we need to find the difference \( m - n \) given that the total number of subsets of the first set (with \( m \) elements) is 48 more than the total number of subsets of the second set (with \( n \) elements).

### Step-by-step Solution:

1. **Understanding the number of subsets**:
The number of subsets of a set with \( p \) elements is given by \( 2^p \). Therefore:
- The number of subsets of the first set (with \( m \) elements) is \( 2^m \).
- The number of subsets of the second set (with \( n \) elements) is \( 2^n \).

2. **Setting up the equation**:
According to the problem, the total number of subsets of the first set is 48 more than that of the second set. This can be expressed mathematically as:
\[ 2^m = 2^n + 48 \]

3. **Rearranging the equation**:
We can rearrange the equation to isolate \( 2^m \):
\[ 2^m - 2^n = 48 \]

4. **Factoring the left-hand side**:
We can factor the left-hand side using the property of exponents:
\[ 2^n(2^{m-n} - 1) = 48 \]

5. **Finding possible values**:
Since \( 2^n \) must be a power of 2, we can test various values of \( n \) to find suitable values for \( 2^n \) that divide 48:
- If \( n = 0 \): \( 2^0 = 1 \) → \( 1(2^{m-0} - 1) = 48 \) → \( 2^m - 1 = 48 \) → \( 2^m = 49 \) (not a power of 2)
- If \( n = 1 \): \( 2^1 = 2 \) → \( 2(2^{m-1} - 1) = 48 \) → \( 2^{m-1} - 1 = 24 \) → \( 2^{m-1} = 25 \) (not a power of 2)
- If \( n = 2 \): \( 2^2 = 4 \) → \( 4(2^{m-2} - 1) = 48 \) → \( 2^{m-2} - 1 = 12 \) → \( 2^{m-2} = 13 \) (not a power of 2)
- If \( n = 3 \): \( 2^3 = 8 \) → \( 8(2^{m-3} - 1) = 48 \) → \( 2^{m-3} - 1 = 6 \) → \( 2^{m-3} = 7 \) (not a power of 2)
- If \( n = 4 \): \( 2^4 = 16 \) → \( 16(2^{m-4} - 1) = 48 \) → \( 2^{m-4} - 1 = 3 \) → \( 2^{m-4} = 4 \) → \( m - 4 = 2 \) → \( m = 6 \)
- If \( n = 5 \): \( 2^5 = 32 \) → \( 32(2^{m-5} - 1) = 48 \) → \( 2^{m-5} - 1 = 1 \) → \( 2^{m-5} = 2 \) → \( m - 5 = 1 \) → \( m = 6 \)

6. **Finding \( m - n \)**:
From the calculations, we found \( m = 6 \) and \( n = 4 \). Therefore:
\[ m - n = 6 - 4 = 2 \]

### Final Answer:
The value of \( m - n \) is \( 2 \).