Yep, many logs on the fire makes for a big fire, and many applications make for a success somewhere along the way! It may not be the one you want that you get, but it's a foot in the door and that's worth a lot when starting out.
I agree, call them and find out, it also lets them know that you really want to go there and work. Don't let them see this as over eagerness however, or they may take advantage in other ways, like reduced salary and benefits etc.
Good luck!
