Anything open-and-shut is straightforward and obvious. Lawyers and detectives love an open-and-shut case because it’s easy to resolve. TV crime shows don’t have a lot of open-and-shut cases because they’re no fun to figure out.
Police detectives working on an open-and-shut bank robbery case have no doubt about who the thief is, and they may even know exactly where he's hiding out. An open-and-shut case of workplace discrimination is easy to prove, and the bosses are going to get in big trouble. An open-and-shut case is so obvious that it doesn't take long to wrap it up and move on to the next one. This American English phrase was first used in 1840s New Orleans.