Question

Let $A$ and $B$ be two sets. The set $A$ has $2016$ more subsets than $B.$ If $A\cap B$ has $3$ members, then the number of members in $A\cup B$ is

• A. $10$
• B. $11$
• C. $12$
• D. $13$

Answer 1

Answer: (D)

If $A$ has $m$ members and $B$ has $n$ members, then
$2^m-2^n=2016=2^5.63$
$\Rightarrow 2^n\left(2^{m-n}-1\right)=2^5\left(64-1\right)=2^5\left(2^6-1\right)$
$\Rightarrow n=5$ and $m-n=6\Rightarrow m=11$
The number of members in $A\cup B$
$=11+5-3=13$