We propose a new way of automating statistical structural testing, based on the combination of uniform generation of combinatorial structures, and of randomized constraint solving techniques. More precisely, we show how to draw test cases which balance the coverage of program structures according to structural testing criteria. The control flow graph is formalized as a combinatorial structure specification. This provides a way of uniformly drawing execution paths which have suitable properties. Once a path has been drawn, the predicate characterizing those inputs which lead to its execution is solved using a constraint solving library. The constraint solver is enriched with powerful heuristics in order to deal with resolution failures and random choice strategies.