Final answer:
The total number of different playlists DJ Bronson can create is calculated by multiplying the permutations of the 7 blues songs with the permutations of the remaining 3 spots chosen from the rock, country, or jazz songs. The final result is then presented in scientific notation.
Step-by-step explanation:
DJ Bronson has 7 blues songs that he wishes to include in his playlist. The remaining 3 spots on his 10-song playlist can be filled with any combination of rock, country, or jazz songs. There are 6 rock, 4 country, and 6 jazz songs available to choose from, making a total of 16 songs for the remaining 3 spots.
To calculate the number of different playlists he can make, we need to find the arrangements for the 7 blues songs in specific spots and then find the combinations of the remaining songs for the 3 spots. The blues songs can be arranged in 7! (factorial) ways. For the remaining 3 spots, since the order in which they are played matters, we use the permutation formula for nPr, which refers to the number of ways of selecting r items (spots on the playlist) from a larger set of n (available songs) without replacement.
The formula for the permutation is:
nPr = n! / (n-r)!
Substituting the values, we have:
16P3 = 16! / (16-3)!
After calculating, we would get the permutations for the 3 remaining spots.
The total number of different playlists possible is the product of the arrangements for the blues songs and the permutations for the remaining spots, presented in scientific notation rounded to the hundredths place.