You're friend tells you about a new online music site called ComboAl-
bum that lets you choose any 10 songs from their song library for 9:99. ComboAlbum's advertising
says their song catalogue is so large there are over 10; 000; 000 diff�erent song combinations available
for download.
1.What is the minimum size of their song catalog if this claim is true?
2.If the song catalogue had the number of songs found in the previous part, but you
could download combinations of 5 songs for 4:99, how many di�fferent combinations of 5 songs would
there be?
3.If for 2:99 ComboAlbum will sell you a randomly selected combination of 10 songs,
what is the probability that your favorite 5 songs in the song library are among the 10 selected.
Again, use the total number of songs found in the �rest part.
let the number of songs be n
C(n,10) > 10 000 000
n!/(10!(n-10)) > 10 000 000
By trial and error, using my C(n,r) key on my calculator
C(25,10) = 3 268 760
C(26,10) = 5 311 735
C(27,10) = 8 436 285
C(28,10) = 13 123 110
1. So they must have at least 28 songs, (not very impressive)
2. Combinations of 5 from those 28
= C(28,5) = 98280
3. prob --- we need 5 of your favourite 5 plus 5 of the remaining 23
prob = C(5,5) x C(23,5)/C(28,10) = .002564
Thank you!
To answer these questions, we need to understand the concept of combinations, which is a way of counting the number of possible selections without regard to the order in which the items are chosen.
1. To find the minimum size of ComboAlbum's song catalog if there are over 10,000,000 different song combinations available, we need to determine the number of songs in the catalog. We can use the concept of combinations to solve this.
The formula for combinations is nCr = n! / (r! * (n-r)!), where "n" represents the total number of elements to choose from, and "r" represents the number of elements being chosen.
In this case, we can use the formula to find the minimum size of the song catalog:
10,000,000 = nCr = n! / (10! * (n-10)!)
We can simplify this equation by multiplying both sides by (10! * (n-10)!):
10,000,000 * (10! * (n-10)!) = n!
By rearranging the equation, we can solve for the minimum value of "n":
n! = 10,000,000 * (10! * (n-10)!)
n! / (10! * (n-10)!) = 10,000,000
The left side of the equation represents the number of combinations when choosing "r" elements from "n" total elements, so the left side of the equation is actually equal to the number of combinations we are looking for, which is 10,000,000.
By trial and error, we can determine that the minimum value for "n" is 15. Therefore, the minimum size of ComboAlbum's song catalog is 15.
2. If the song catalog has 15 songs, and combinations of 5 songs can be downloaded for $4.99, we need to calculate the number of different combinations of 5 songs.
Again, we can use the formula for combinations:
nCr = n! / (r! * (n-r)!)
In this case, n = 15 (the number of songs in the catalog) and r = 5 (the number of songs in each combination). Plugging these values into the formula, we get:
15C5 = 15! / (5! * (15-5)!)
Calculating this, we find that there are 3,003 different combinations of 5 songs available.
3. If ComboAlbum sells a randomly selected combination of 10 songs for $2.99, we need to calculate the probability that your favorite 5 songs are among the 10 selected.
The probability can be calculated by dividing the number of favorable outcomes (the number of combinations that include your favorite 5 songs) by the total number of possible outcomes (the total number of combinations of 10 songs).
Since the previous part determined that there are 3,003 different combinations of 5 songs, the number of favorable outcomes is also 3,003.
To calculate the total number of possible outcomes, we need to calculate the number of combinations of 10 songs from the 15-song catalog:
15C10 = 15! / (10! * (15-10)!)
Calculating this, we find that there are 3,003 different combinations of 10 songs.
Therefore, the probability of your favorite 5 songs being among the 10 selected is 3,003/3,003, which simplifies to 1 or 100%.