Question: Let $S=\{1,2, \cdots, 10 \}$. Then the number of pairs $(A, B)$, where $A$ and $B$ are non-empty disjoint subsets of $S$ is?
[I could solve the question as demonstrated below, but it involves calculating a tedious sum of products, which would take up a lot of time. It is typically expected to solve this in a couple of minutes, so I was wondering if there was a faster way to do this.]
My approach: Let the set $A$ consist of $x$ elements. There are ${10 \choose x}$ ways of making that selection.
We are now left with $10-x$ elements.
Let the set $B$ consist of $y$ elements. This selection can be done by ${10-x \choose y}$ ways.
The total number of ways can be found out by summing over the product of the two above as $\sum_{x=1}^{9} \sum_{y=1}^{10-x} {10 \choose x}{10-x \choose y} $ which comes out to be $57002$