Question

Let $X=1,2,3,.....,11$. Find the the number of pairs {A, B} such that $A\subseteq X$. $B\subseteq X$. $A\ne B$ and $A\cap B=4,5,7,8,9,10$.

Answer 1

Let $A\cup B=Y,B\setminus A=M,A\setminus B=N$ and $X\setminus Y=L$, then X is the disjoint union of M, N, L and $A\cap B$.
We have, $A\cap B=4,5,7,8,9,10$ is fixed. The remaining 5 elements 1, 2, 3, 6, 11 can be distributed in any of the remaining sets M, N, L.
This can be done in $3^5$ ways.
Of these if all the elements are in the set L, then $A=B=4,5,7,8,9,10$ and this case has to be omitted.
Hence the total number of pairs {A, B} such that $A\subseteq X,B\subseteq X,A\ne B$ and $A\cap B$ = {4, 5, 7, 8, 9, 10} is $3^5-1$.