A music lover has 28 songs and is able to make a playlist of 12 songs (order doesn't matter). How many different 12-song playlists can be made? Is it realistic and practical to make all of them?

1 Answer

Answer:

30,421,755 CDs. Realistic to make a CD of every combination? No. Practical? What is practicality to a true music lover?!?!?!

Explanation:

We have 28 songs and we're going to find how many ways we can put sets of 12 songs onto CDs. Since order doesn't matter (having songs 1 - 12 on a CD is the same as having songs 12 - 1 on the CD), we'll use the Combination calculation, which is:

#C_(28, 12)=(28!)/((12!)(28-12)!)=(28!)/((12!)(16!)#

And now let's evaluate this fraction:

#(28xx27xx26xx25xx24xx23xx22xx21xx20xx19xx18xx17xxcancel(16!))/((12!)cancel(16!)#

#(28xx27xx26xx25xx24xx23xx22xx21xx20xx19xx18xx17)/(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2)#

There's a lot of cancelations, so I'll use colours to help keep track:

#(cancel(color(red)28)xxcancel(color(blue)(27))xx26xxcancel(color(orange)(25))^5xxcancel(color(green)(24))xx23xxcancel(color(pink)(22))^2xx21xxcancel(color(brown)(20))^2xx19xxcancel(color(tan)(18))^3xx17)/(cancel(color(green)(12))xxcancel(color(pink)(11))xxcancel(color(brown)(10))xxcancel(color(blue)(9))xx8xxcancel(color(red)(7))xxcancel(color(tan)(6))xxcancel(color(orange)(5))xxcancel(color(red)4)xxcancel(color(blue)3)xxcancel(color(green)(2))#

Let's clean this up and do one more round... I'm going to express 26 as #13xx2#

#(13xxcancel2xx5xx23xxcancel2xx21xxcancel2xx19xx3xx17)/cancel8#

#(13xx5xx23xx21xx19xx3xx17)=30,421,755#

(To avoid the long and involved calculations, you can use the Combination calculator tool here: http://www.calculatorsoup.com/calculators/discretemathematics/combinations.php)