That formula, is the subject of the so-called (*) BEST theorem and gives the number of unicursal trails named for this scientist.
Matroids named after this man have a rank that, when incremented by one, is divisible by the cardinality of the underlying set. Hierholzer's algorithm and Fleury's algorithm compute structures  named for this scientist. The number of those structures named for this scientist is given by the following formula: the number of arborescences times the product of "the degree of each vertex in the graph minus one" factorial. That formula, is the subject of the so-called (*) BEST theorem and gives the number of unicursal trails named for this scientist. This mathematician proved that an even number of vertices will have an odd degree in all undirected graphs. That result, the handshaking lemma, was proved by him when he discussed namesake tours in which every edge of a graph is visited exactly once. The Seven Bridges of Konigsberg problem was solved by, for 10 points, what Swiss mathematician?
Matroids named after this man have a rank that, when incremented by one, is divisible by the cardinality of the underlying set.
Leonhard Euler 