# Can you make an infinite number of playlists?

When mowing the lawn, I like to listen to podcasts. One of my favorites is [Buzz Out Loud](http://bol.cnet.com). This weekend, I was listening to episode 817 and one of the topics of discussion was MySpace and their DRM free music stuff. [Wired](http://blog.wired.com/music/2008/09/myspace-launche.html) had a description of what they were going to do. That is not my point. The point is the claim that you could make an infinite number of playlists. How about I calculate (or estimate) the number of different playlists one could make.

First, the idea behind the idea. Calculating the number of combinations is not my strong suit. I always get myself confused. So, if I make an error, feel free to point it out. [Wikipedia](http://en.wikipedia.org/wiki/Combinatorics) calls it combinatorics. Not sure if that is the real name for this stuff, but I guess it doesn’t matter.

Ok. Now for the parameters. The Wired story says that MySpace will allow playlists up to 100 songs. How many songs does one have to choose from? If it were my music collection, I have 2107 songs. I suspect that this is below average (the reason for this suspicion is that I rarely acquire new music so I figure my library is smaller than the norm). What about iTunes? How many songs are available on that? According to [wikipedia](http://en.wikipedia.org/wiki/ITunes_Store) there are 8 million songs you can pick from.

The first approximation to this answer then would be:

*How many unique sets of 100 can you create from 8 million songs?*

I will assume that order matters – it does, doesn’t it? I mean if I make a playlist with Michael Jackson’s Thriller followed by Van Halen’s Jump, that is different than Jump followed by Thriller – right? Also note that I am calculating the number of ways you can make songs with a 100 playlists out of 8 million. You could also make a playlist of 99 songs, or 98.

How many combinations can you make of *n* choices out of *b* possible to choose from? The easiest way to remember this is to think about something like a combination lock. If I have 3 10 digit number for my combination, how many are there? Well, there is 000, 001, 002, 003, ….999 (or 1000 combinations). For that example I chose 3 out of 10 for a possible number of 103 = 1000 combinations. That means:

Now applying this to “infinite playlists”, here

This would give the possible combinations as:

That is a big number. I think you could argue that is close enough to infinity. You could also argue that is NOT infinity (it depends on your agenda). But is this even a good estimate? What if I only wanted 99 songs in my playlist? Do I need to count those as well? If I repeat the above calculation with only 99 songs in the playlist, I get:

From this, you can see that I do Not need to consider non-100 song playlists. The number of combinations for 100 songs is 10 million times more than that for 99 songs.

There is one weak part of my argument. I have assumed that you can use the same song more than once. In fact, this calculation includes the possibility of a playlist that consists of 100 instances of “Love me Tender” by Elvis. Ok, that would be silly. Why would anyone do that? They could just have a playlist of 1 “Love me Tender” and put it on repeat. So There are 8 x 106 possible playlists with 100 of the same song that should not have been counted. BUT! There are also 8 x 106 playlists with just 1 song that I did not count. So, it evens out. I guess my original answer is ok.

I think one could still argue whether this is “infinity” or not. I will stay out of that battle.