DJ Jacqueline is making a playlist for an mp3 player; she is trying to decide what
12
12
songs to play and in what order they should be played. If she has her choices narrowed down to
21
blues,
13
reggae, and
25
hip-hop songs, and she wants to play an equal number of blues, reggae, and hip-hop songs, how many different playlists are possible? Express your answer in scientific notation rounding to the hundredths place.
So, you want 4 of each kind.
Pick any set of 4 from each kind, and then there will be 12! ways to play them.
21C4 * 13C4 * 25C4 * 12! = 2.593*10^19
To determine the number of different playlists possible, we need to use the concept of permutations. Since DJ Jacqueline wants to play an equal number of blues, reggae, and hip-hop songs, she needs to choose 12 songs from each genre.
The number of ways to choose 12 songs from 21 blues is denoted as "21C12" or "21 choose 12," which can be calculated using the formula:
21C12 = (21!)/(12!(21-12)!) = (21!)/(12!9!)
Similarly, the number of ways to choose 12 songs from 13 reggae and 12 songs from 25 hip-hop can be calculated as:
13C12 = (13!)/(12!(13-12)!) = (13!)/(12!1!)
25C12 = (25!)/(12!(25-12)!) = (25!)/(12!13!)
To find the total number of possible playlists, we need to multiply these three combinations together:
Total number of playlists = (21C12) * (13C12) * (25C12)
Let's calculate this:
(21C12) = (21!)/(12!9!) = 116,280
(13C12) = (13!)/(12!1!) = 13
(25C12) = (25!)/(12!13!) = 11,690
Total number of playlists = 116,280 * 13 * 11,690 ≈ 1.99 x 10^9
Therefore, the number of different playlists possible is approximately 1.99 x 10^9 (rounded to the hundredths place) in scientific notation.
To find the number of different playlists, we need to determine the number of ways we can select an equal number of blues, reggae, and hip-hop songs.
First, let's find the smallest number of songs among the three genres. In this case, it's the blues songs with a total of 21.
Since we need to play an equal number of blues, reggae, and hip-hop songs, we can choose from 0 through 21 songs from the blues genre. Let's consider the number of songs from the blues genre as "x."
Now, the number of songs from the other genres (reggae and hip-hop) would also be equal to "x" in order to maintain an equal number of songs.
Therefore, the number of different playlists can be calculated by finding the number of different values for "x" ranging from 0 to 21. Since the values of "x" are discrete (integer values), the count of different playlists would be (21 + 1) = 22.
Now, to express this answer in scientific notation and rounding to the hundredths place, we have:
The number of different playlists = 22
Scientific Notation: 2.20 x 10^1
Rounded to the hundredths place: 2.20
So, there are 2.20 (rounded to the hundredths place) different playlists possible for DJ Jacqueline.