In the Land of Wonders, $n$ cities are connected by roads (at most one road connects two cities). There are a total of $a$ roads. It is assumed that, if two cities are directly connected by a road, there always exists an indirect path (i.e., a sequence of roads) to go from one to the other. The Queen of Hearts has decided that henceforth an even number of roads will depart from each city, and her minister proposes a list of all possible systems, totaling $r$, obtained by removing some of the existing roads. What is the total number of roads appearing on this list (counting each road as many times as it appears)?
Let $e$ be a road, and let $C$ be a cycle containing it, which exists by assumption. We obtain a natural bijection between the systems containing $e$ and those not containing $e$ by inverting all the roads of $C$. Therefore, $e$ is in exactly half of the systems, and the answer is $\frac{n r}{2}$.