Let $P_1=\{A_1,A_2,...,A_{10}\}$ and $P_2=\{B_1,B_2,...,B_{10}\}$ be two distinct partitions of a set of $100$ people into groups of $10$. Let $G$ be a bipartite graph with $P_1$ and $P_2$ as bipartitions. There is an edge between $A_i$ and $B_j$ if $A_i$ and $B_j$ are disjoint.
Show $G$ has a matching.