Let A and B be two finite sets with m and n elements respectively. The total number of subsets of the set A…

We know that, the total number of subsets of a set having $n$ elements is $2^n$.

$\Rightarrow 2^{\mathrm{m}}-2^{\mathrm{n}}=56$

$\Rightarrow 2^{\mathrm{n}}\left(2^{\mathrm{m}-\mathrm{n}}-1\right)=2^3\times 7$

$\Rightarrow 2^{\mathrm{n}}=2^3$ and $2^{\mathrm{m}-\mathrm{n}}-1=7$

$\Rightarrow \mathrm{n}=3$  and $2^{\mathrm{m}-\mathrm{n}}=8$

$\Rightarrow \mathrm{n}=3$ and $\mathrm{m}-\mathrm{n}=3$

$\Rightarrow \mathrm{n}=3$ and $\mathrm{m}=6$

Distance between $\mathrm{P}\left(6, 3\right)$ and $\mathrm{Q}\left(-2,-3\right)$ is given by,

$\mathrm{PQ}=\sqrt{{\left(6+2\right)}^2+{\left(3+3\right)}^2}$

$\Rightarrow \mathrm{PQ}=\sqrt{8^2+6^2}$

$\Rightarrow \mathrm{PQ}=\sqrt{100}$

$\Rightarrow \mathrm{PQ}=10$ units