Bipartite Graphs and Matchings · Hard problem

Five applicants $P,Q,R,S,T$ apply for five openings $1,2,3,4,5$. $P,Q,R$ are each qualified only for $\{1,2\}$; $S$ is qualified for $\{3,4\}$; $T$ is qualified for $\{4,5\}$. What is the largest number of applicants who can each be placed in a distinct opening they are qualified for?