Putnam 1990 A6

If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S, T)$ of subsets of $\{1, 2, \dots, n\}$ admissible if $s > |T|$ for each $s \in S$, and $t > |S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1, 2, \dots, 10\}$ are there? Prove your answer.