Number of perfect matchings in a complete bipartite graph without any edge repetition
We can reason this way. After we have found another perfect matching, we remove all the edges of this matching from our graph. We will obtain a new regular bipartite graph for which we can use this statement. After that we repeat the above process. It is clear that the process will end when all the edges are exhausted. As a result, we will get exactly $n/2$ of perfect matchings which have no edges repeated.