How hard is it to get a job within the LEGO company, straight out of university as a graduate? How hard is it to get a job at the LEGO company?

Being fresh out of university as a graduate of mathematics, I'm specifically interested in answers on if and how LEGO hires graduates.
LEGO has a list of open positions which you can consult. As far as I understand, the application process starts there and you have to register a profile to apply for positions. Some positions also list a direct point of contact who can answer questions on the specific position (usually the boss-to-be, I suppose).

I'm not sure where a mathematics background might be useful within LEGO, but I don't know the full organization by far, and there are certainly some positions where it's going to be useful. In any case, don't be too afraid by all requirements listed for given positions, they don't necessarily mean that someone isn't suited for the job. But that's probably a discussion for another site.

Of course, jobs for which experience is required will probably be out of your reach, unless you do have experience in the required domain. For example, you could very well be a good LEGO builder already and have a chance for a design job.

But the good news is that LEGO seems to be willing to attract young talents, in particular in the IT sector. I remember seeing recently a "job position" which was in fact some sort of one-day seminar for young graduates, so that they could see what the life at LEGO is. There are also open positions which specifically mention students.

In any case, enthusiasm about the LEGO brand is certainly useful, and of course you'll probably have to devote some thinking time about where you're willing to work and live, especially considering most central positions are in specific countries, with Denmark being the most important.