Name: Linda

Who is asking: Teacher

Level: Middle

Question:

Your DJ Business has 6 rap, 10 rock, 6 alternative, 8 oldies, and 5 country CD singles. How many different 10-song sets can the DJ play, if she plays 2 singles from each category? and How many different 10-song sets if she plays exactly 3 rap singles and 4 rock singles in each set?

Hi Linda,

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 _{6}C_{2} ("6 choose 2"). Similarly you can choose _{10}C_{2} rock songs, _{6}C_{2} alternative, _{8}C_{2} oldies and _{5}C_{2} country songs.

The next step is to apply the following counting principle:

If there are m ways in which to perform one action, and n ways in which to perform a second action, then there are mn ways the two actions can be performed
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,

Leeanne