Question

Let $A$ and $B$ be two finite sets. The set $A$ has $480$ more subsets than $B.$ If $A\cap B$ has $2$ elements, then the number of elements in $A\cup B$ is

• A. $14$
• B. $13$
• C. $12$
• D. $10$

Answer 1

The correct option is C $12$

Let $A$ has $m$ elements and $B$ has $n$ elements. $(m>n)$
Then, $2^m-2^n=480$
$\Rightarrow 2^n(2^{m-n}-1)=32\times 15=2^5(2^4-1)$
$\Rightarrow n=5$ and $m-n=4$
$\Rightarrow n=5$ and $m=9$

$\therefore n\left(A\right)=9$ and $n\left(B\right)=5$
$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$
$=9+5-2=12$