Question

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m, n)$ from the point $Q(-2, -3)$ is:

• A. 10
• B. 6
• C. 4
• D. 8

Answer 1

Answer: (A)

10

Answer 2

The total number of subsets of a set with $m$ elements is $2^m$ and for a set with $n$ elements is $2^n$. Given:
$2^m = 2^n + 56.$

Rearranging:
$2^m - 2^n = 56.$

Factoring the left side:
$2^n (2^{m-n} - 1) = 56.$

Since $56 = 2^3 \times 7$, we set $2^n = 8 \implies n = 3$ and
$2^{m-n} - 1 = 7 \implies 2^{m-n} = 8 \implies m - n = 3.$
Therefore:
$m = 6, \quad n = 3.$

The distance between points $P(6, 3)$ and $Q(-2, -3)$ is given by:

$\text{Distance} = \sqrt{(6 - (-2))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10.$

Thus, the correct answer is 10.