Question

If the difference between the number of subsets of two finite sets $A$ and $B$ is $120,$ then $n(A\times B)$ is

• A. $21$
• B. $24$
• C. $25$
• D. $120$

Answer 1

The correct option is A $21$

Let $n\left(A\right)=\alpha$ and $n\left(B\right)=\beta$ with $\alpha >\beta$
Given that difference between the number of subsets of sets $A$ and $B$ is $120.$
$\therefore 2^{\alpha }-2^{\beta }=120$
$\Rightarrow 2^{\beta }(2^{\alpha -\beta }-1)=8\times 15$
Comparing both sides, we get
$8=2^{\beta }\text{and}15=2^{\alpha -\beta }-1$
$\Rightarrow 2^3=2^{\beta }\text{and}{2}^4=2^{\alpha -\beta }$
$\Rightarrow \beta =3\text{and}4=\alpha -\beta$
$\Rightarrow \alpha =4+\beta =7$
$\Rightarrow (\alpha ,\beta )=(7,3)$
$\therefore n(A\times B)=n\left(A\right)\times n\left(B\right)=21$