The best advice I ever got was: There's more to the games industry than code, art, and music. I got in via marketing & PR and have been a pro in the industry for over a decade now.
That is true advice, there are more options available.
A studio needs many different jobs than just those.
A given project may have 20 programmers and their leads, 30 art folk and their leads, 1-3 producers, 1-3 designers, a few audio people that are shared with other projects, and shared other groups for IT, HR, legal, studio management, business relations, marketing and other stuff. QA comes and go through contracts. The approximate ratio scales up and down for large and small projects. By far the most reliable way to enter is through programming and art fields.
Seeing as the original post wrote "Just completed my degree in CS and wondering what to do next." then the programming track seems the most reasonable one in this case.