Hello, I've been struggling with this exercise: Let [itex]X[/itex] be set with [itex]n^2+n+1[/itex] elements and let [itex]S[/itex] be system of [itex](n+1)[/itex]-sized subsets of [itex]X[/itex] such that every two sets in [itex]S[/itex] have at most one common element and [itex]|S| = n^2 + n + 1[/itex]. Prove that S has a system of distinct representatives. Of course this is an exercise for use of Hall's theorem (aka Marriage theorem). I tried it by induction of index set ([itex]J[/itex] in Hall's theorem) and somehow directly as well, but I still can't prove it. Thank you for any advices.