Partitioning Sets of Quintuples into Designs

It is shown that the collection of all \(\dbinom{12}{5}\) quintuples chosen from a set of twelve points can be partitioned into twelve mutually disjoint \(4-(11,5,1)\) designs in precisely \(24\) non-isomorphic ways. The results obtained are then used to show that the collection of all \(\dbinom{13}{6}\) hextuples chosen from a set of thirteen points cannot be partitioned into thirteen mutually disjoint \(5-(12,6,1)\) designs.