We actually have nearer 200 people working here, with around 160 peole working on Kinect Disneyland Adventures. The design team is about 20 people strong. Thinking about it, my 50% estimation was a bit overely optimistic, it's more like 40% - but still that's a good few designers. A lot of whom are Seniors.
Edit: I'll also add that if I were personally looking to hire a designer; I wouldn't filter applicants by if they have a degree or not. If their portfolio can demonstrate their understanding of game design and I can see they can think like a designer, I'd have no issue hiring them.
I'm going to re-post something I posted a while ago, as it's still my opinion about all this:
Right, here's some advice:
1: If you're dead set on being a designer, then no - you don't really need a degree to get a job designing games. Your portfolio should be enough to get you an interview, then it's really all down to how well you interview. If you're socially inept, then you may struggle to land a gig.
1a: Your portfolio should consist of actual playable maps in other games, sketches of those maps with annotations explaining why you did things the way you did. Write down what you expect the player to experience; you're trying to get across how your mind works in as little words a possible. Sample documentation to show your writing ability is also a plus.
1b: Get a job in the industry. Doesn't matter what you're doing - if you're inside the industry you have a better chance than being outside of it. Specifically, look at developers and see if their QA departments are hiring. So many designers and production staff come from internal QA (generally towards the end of a project when a member of staff leaves and they need someone to jump in and finish off what they were doing), so see if you can land a gig there.
2: If you're not absolutely certain you want to be a designer, then go take a degree in something else that interests you - get a skill outside of games and then try and re-enter the industry. If it doesn't work out, then go back to your other trade. Creative writing degree maybe?
I'm absolutely certain that if you put the effort in and work hard, you'll get to where you want to be. A lot of people want an easy way in, but there isn't an easy way. You have to work hard and show your worth - it's a very competitive industry!