DJ Lorraine is making a playlist for a radio show; she is trying to decide what 15 songs to play and in what order they should be played. If she has her choices narrowed down to 19 rock, 14 pop, and 21 blues songs, and she wants to play an equal number of rock, pop, and blues songs, how many different playlists are possible? Express your answer in scientific notation rounding to the hundredths place.

6.19 * 10^23 playlists

Step-by-step explanation:

She needs to decide 15 songs, and the number of songs of each of the 3 styles is the same, so we have 5 songs for each style.

The number of different groups of 5 rocks songs she can choose is a combination of 19 choose 5:

C(19,5) = 19! / (5! * 14!) = 19*18*17*16*15/(5*4*3*2) = 11628

In the same way, the number of different groups of 5 pop songs she can choose is a combination of 14 choose 5:

C(14,5) = 14! / (5! * 9!) = 14*13*12*11*10/(5*4*3*2) = 2002

The number of different groups of 5 blues songs she can choose is a combination of 21 choose 5:

C(21,5) = 21! / (5! * 16!) = 21*20*19*18*17/(5*4*3*2) = 20349

Then, the order of each song is important, so the number of possibilities for sorting 15 songs is calculated by the factorial of 15:

15! = 1,307,674,368,000

So the number of possible playlists is the product of each number of different groups and the sorting possibilities:

Number of playlists = 11628 * 2002 * 20349 * 1307674368000

Number of playlists = 6.19 * 10^23

the answer is d

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