Answer:
5878600 different playlists.
Step-by-step explanation:
Since we require 14 songs for the playlist and there are 7 hip-hop, 5 jazz, 7 pop, and 8 blues songs, and DJ Mykel wants to play no more than 3 hip-hop songs, the number of ways in which he can combine the 7 hip-hop songs to play no more than 3 is ⁷C₃ = 7!/4!3!.
Now, since we have 3 songs already selected, we are left with 14 - 3 = 11 songs to select.
Since we have selected the 3 hip-hop songs, we are left with 5 jazz, 7 pop, and 8 blues songs which sum to a total of 20 songs.
So, we have 20 songs to select in 11 ways which is ²⁰C₁₁ = 20!/(20 - 11)!11! = 20!/9!11!
So, the number of different playlists that can be formed is thus
⁷C₃ × ²⁰C₁₁ = 7!/4!3!.× 20!/9!11!
= 35 × 167960
= 5878600 different playlists.
