Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subse

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\

Professor PC Thomas Classes · Answer

(c) : Total number of subsets of set\ $A=2^{m}$\ Total number of subsets of set $B=2^{n}$\ According to question, $2^{m}-2^{n}=56$\ $\Rightarrow \quad 2^{n}\left(2^{m-n}-1ight)=2^{3} 	imes 7$\ $\Rightarrow \quad 2^{n}=2^{3}$ and $2^{m-n}=8=2^{3}$\ $\Rightarrow \quad n=3$ and $m-n=3$\ $\Rightarrow \quad m=3+n=3+3=6$\ $	herefore \quad m=6$ and $n=3$ Now, distance between the point $P(6,3)$ and $Q(-2,-3)$ is given by\ $P Q=\sqrt{(6+2)^{2}+(3+3)^{2}}=\sqrt{64+36}=10$ units\