Thinking About Thinking

I was thinking randomly about how many bit.ly combinations there are and how much unique shortened URLs they can generate given their current construct.  Here’s their construct:

• Up to 6 characters after bit.ly/
• Any mix of capitalized letters, lowercase letters, and numbers

If you limit the combinations to only ones where bit.ly is the domain (and not partner domains), and you presume they can go up to 6 characters (i.e. 1–6) as opposed to exactly 6 characters – how many total combinations are there?  Any math whiz out there who can figure it out pretty quickly and enlighten us in the comments section?  My small math brain was getting cramps thinking about it. 

Given that they shorten 40–50 million URLS per day, I also wonder how long they can last on this construct.  Probably a long time I suspect even including their growth.  Random thought for the day, and thanks for the help.