Difference of subsets of 2 finite sets with m & n elements, is 56 then find distance between (m, n) & (-2, -3).

Options

A. 10
B. 16
C. 14
D. None of these

Answer

To solve the problem step by step, we need to find the values of \( m \) and \( n \) based on the information given about the subsets of two finite sets, and then calculate the distance between the points \( (m, n) \) and \( (-2, -3) \).

### Step 1: Set up the equation based on the number of subsets
The number of subsets of a set with \( m \) elements is given by \( 2^m \) and for a set with \( n \) elements, it is \( 2^n \). According to the problem, the difference in the number of subsets is 56:
\[ 2^m - 2^n = 56 \]

### Step 2: Factor the equation
We can factor the left-hand side:
\[ 2^n(2^{m-n} - 1) = 56 \]
This means \( 2^n \) must be a factor of 56. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

### Step 3: Determine possible values for \( n \)
Since \( 2^n \) must be a power of 2, the possible values for \( n \) are:
- \( n = 0 \) (then \( 2^n = 1 \))
- \( n = 1 \) (then \( 2^n = 2 \))
- \( n = 2 \) (then \( 2^n = 4 \))
- \( n = 3 \) (then \( 2^n = 8 \))
- \( n = 4 \) (then \( 2^n = 16 \))
- \( n = 5 \) (then \( 2^n = 32 \))
- \( n = 6 \) (then \( 2^n = 64 \))

### Step 4: Test possible values of \( n \)
We will test these values to find \( m \):
1. If \( n = 3 \):
\[ 2^3(2^{m-3} - 1) = 56 \implies 8(2^{m-3} - 1) = 56 \implies 2^{m-3} - 1 = 7 \implies 2^{m-3} = 8 \implies m - 3 = 3 \implies m = 6 \]
This gives us \( m = 6 \) and \( n = 3 \).

### Step 5: Calculate the distance between the points
Now we need to find the distance between the points \( (m, n) = (6, 3) \) and \( (-2, -3) \). The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting our points:
\[ d = \sqrt{((-2) - 6)^2 + ((-3) - 3)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Final Answer
The distance between the points \( (m, n) \) and \( (-2, -3) \) is \( 10 \).