# Chris has 25 songs in his library and wants to put 12 of them into a playlist. If he picks songs at random, how many different groups of songs can be created?

##### 1 Answer

$5 , 200 , 300$ different playlists (and at this point we're only grouping songs - we aren't even playing with different ordering of playlists!)

#### Explanation:

We're working with a combination, which means a playlist that includes songs A and B, no matter the order, are the same (a permutation, on the other hand, counts playlists with different song orders as different).

The general equation for a combination is:

C_(n,k)=(n!)/((k)!(n-k)!) with $n = \text{population", k="picks}$

C_(25,12)=(25!)/((12)!(25-12)!)=(25!)/((12!)(13!))

There are two ways to evaluate this - we can choose to go the multiplication route:

(25xx24xx23xx22xx21xx20xx19xx18xx17xx16xx15xx14xx13!)/(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xx13!)

and work through that cancellation nightmare (but do-able on a simple calculator), or we can recognize that factorials are really just numbers (ex. 3! = 6, 4! = 24, etc) and so substitute in using a factorial table:

$\frac{15 , 511 , 210 , 043 , 330 , 985 , 984 , 000 , 000}{\left(479 , 001 , 600\right) \left(6 , 227 , 020 , 800\right)}$

Since my calculator doesn't stand a chance of working this through, I'll copy and paste into google calculator:

and we get $5 , 200 , 300$