Let $X$ be a set of $5$ elements.The number $d$ of ordered pairs $(A,B)$ of subsets of $X$ such that $A$ not equal to $\phi$, $B$ not equal to $\phi$, $A \cap B$ not equal to $\phi$ is----
The question implies that $A$ and $B$ must be disjoint.the total number of subsets are $31$.the number of subsets containing 2 elements are $\binom 52.3$Similarly I could figure out the rest of the subsets.But that turns out to be a very large number and it is wrong.
Any help appreciated.Thanks.
There are $\binom{5}{k}$ subsets of size $k$. If we choose $A$ to be one of these subsets, then there are $5-k$ other elements that can belong to $B$. There are $2^{5-k}-1$ nonempty subsets of these $5-k$ elements. Summing over $k=1,2,\ldots,4$ gives
$$d = \binom{5}{1} (2^4-1) + \binom{5}{2} (2^3-1) + \binom{5}{3} (2^2-1) + \binom{5}{4} (2^1-1).$$