Here's the skinny: I love math. I think math is fun and exciting and that it teaches you to think analytically. I also love programming. The ability to create systems and apps that no one before me has been able to do is also a very exciting and empowering feelings.
Here's the problem: I think that teaching myself higher order mathematics would be a whole lot harder than teaching myself anything that I could learn in a Computer Science program. From this I reason that I should major in Applied Math (with some Pure Math courses) and teach myself all the programming and development I care to learn. However, I plan on engineering software for a living.
Here's the question(s): I believe that once I get an interview I will be prepared to blow minds, but of course the problem would be getting the interview. Will companies throw my resume in the NO pile because I don't have a Computer Science degree? Also, would I be at a disadvantage when it comes to getting an internship? I strongly believe that a good internship prior to graduation is something that all undergrads should take on and I would not be very happy if I couldn't get one myself.