Who is asking: Teacher
For this question I would take it in steps. For example, if the DJ needs to play 2 rap songs and there are six to choose from, how many ways are there to choose 2 from the 6. The answer is 6C2 ("6 choose 2"). Similarly you can choose 10C2 rock songs, 6C2 alternative, 8C2 oldies and 5C2 country songs.
The next step is to apply the following counting principle:
By generalizing this rule for the case when there are more than just 2 actions (as in this case where there are five), we find that multiplying the indivudual results from step 1 together will yield the answer you are looking for.
Hope this helps,