I think there is a small chance you can get into any distinguished degree program, but it's very rare without building the skill. I thought this was the case when I sent a new Sim with zero skills who I hadn't played yet and I was surprised to see one distinguished degree option pop up for her, I can't remember which one. That's only happened to me twice with a zero or low skilled Sim though. Usually, they get only regular degree options. It's nice that those two Sims got a lucky chance at a good degree because I wasn't going to build their skills up, I just wanted to add them as students lol.
Does anyone know if your traits boost your chances at all?
The residency scholarship though, that's got to be totally random. My Sims who live in the same house don't all get that one. :/
