# 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?

#### 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!)

$\frac{28 \times 27 \times 26 \times 25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 \times 17}{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2}$

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 $13 \times 2$

$\frac{13 \times \cancel{2} \times 5 \times 23 \times \cancel{2} \times 21 \times \cancel{2} \times 19 \times 3 \times 17}{\cancel{8}}$

$\left(13 \times 5 \times 23 \times 21 \times 19 \times 3 \times 17\right) = 30 , 421 , 755$

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