HMMT 2022 U33

In last year's HMMT Spring competition, 557 students submitted at least one answer to each of the three individual tests. Let $S$ be the set of these students, and let $P$ be the set containing the 30 problems on the individual tests. Estimate $A$, the number of subsets $R \subseteq P$ for which some student in $S$ answered the questions in $R$ correctly but no others. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-\frac{2}{3}|A-E|\right\rfloor\right)$ points.