Question 11
Let $A=\{1,2,\ldots,10\}$. Count the number of ordered pairs $(S_1,S_2)$, where $S_1$ and $S_2$ are non-intersecting and non-empty subsets of $A$ such that the largest number in $S_1$ is smaller than the smallest number in $S_2$. For example, if $S_1=\{1,4\}$ and $S_2=\{5,6,7\}$, then $(S_1,S_2)$ is such an ordered pair.