Question

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?

Answer 1

`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="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:

`(15,511,210,043,330,985,984,000,000)/((479,001,600)(6,227,020,800))`

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`