Question

Let $X=\{1,2,3,........,10\}$. Find the the number of pairs $(A,B)$ such that $A\subseteq X$, $B\subseteq X$. $A\ne B$ and $A\cap B=\{2,3,5,7\}$.

Answer 1

Let $A\cup B=Y$, $B$\$A=M$, $A$\$B=N$ and $X$ \$Y=L$.
Then X is the disjoint union of M, N, L and $A\cap B$.
Now $A\cap B$ $=\{2,3,5,7\}$ is fixed.
The remaining six elements $1,4,6,8,9,10$ can be distributed in any of the remaining sets $M,N,L$.
This can be done in $3^6$ ways. Of these if all the elements are in the set L, then $A=B=\{2,3,5,7\}$ and which this case has to be deleted.
Hence the total number of pairs $(A,B)$ such that $A\subseteq X$, $B\subseteq X$. $A\ne B$ and $A\cap B$ $=\{2,3,5,7\}$ is $3^6-1$.