Just so you know, "game design", as I've heard it used by my (numerous) friends in the industry, is not the software engineering part. Designers, like my housemate, are responsible for the game content - the story, the characters, the mechanics, etc.
If you want to get into the industry, there are numerous paths you can take. A business major with some CS or industry-specific classes can be a producer. An art major can be a game artist. A computer science/software engineering major can be a game developer. An electrical engineer can be a console designer. A lit/cultural studies/comparative media/writing major (or for that matter, a CS major with one of these as a minor) can be a game designer. One can probably major in anything and be a tester, as long as one is really, really good at playing video games.