Respuesta :
Since the DJ doesn't want to play more than 4 Reggae songs he can chooses 0, 1, 2, 3 or 4 reggae songs.
Case 1. The DJ chooses 0 Raggae song.
In this case all the songs are selected from the 19 possible songs of different genre, using combinations defined by:
[tex]_nC_k=\frac{n!}{k!(n-k)!}[/tex]We have a total of:
[tex]_{19}C_{10}\cdot_7C_0=92378[/tex]Case 2. The DJ chooses 1 Raggae song.
In this case 9 songs are selected from the 19 songs of other genre then the total possibilities are:
[tex]_{19}C_9\cdot_7C_1=646646[/tex]Case 3. The DJ chooses 2 Raggae songs.
In this case 8 songs are selected from the 19 songs of other genre then the total possibilities are:
[tex]_{19}C_8\cdot_7C_2=1587222[/tex]Case 4. The DJ chooses 3 Raggae songs.
In this case 7 songs are selected from the 19 songs of other genre then the total possibilities are:
[tex]_{19}C_7\cdot_7C_3=1763580[/tex]Case 5. The DJ chooses 4 Raggae songs.
In this case 6 songs are selected from the 19 songs of other genre then the total possibilities are:
[tex]_{19}C_6\cdot_7C_4=949620[/tex]So the total possible ways of choosing the playlist is 5039446.
Now the playlist can be arrange in 10! different ways, hence the are different playlists:
[tex]10!\cdot5039446=1.83\times10^{13}[/tex]