Solving Problems Using Counting Principles

Question

 

Jeanine is selecting various albums to be displayed in the front of her music coffee house. She will be choosing 4 albums for one part of the display. This part of the display is very prominent and she is really trying to promote jazz. There are five albums from Diana Kroll, seven albums from Harry Connick Jr. and four albums from Michael Bublé.  In how many ways can she choose four albums for display and at least two albums are from Diana Kroll? Evaluate using both the direct and indirect method.

Answer 1

Separate the Diana Kroll (DK) albums from the others. There are 10 ways (5*4/2) of selecting two DK albums out of 5. If the DK albums are represented by the letters A to E then the combinations are:
AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.
If all the other albums are represented by letters a to k, then DK combined with other artistes is represented by ABab, ABac,...,ABjk, ACab, ACac,...,...,DEjk.
There are 11*10/2=55 ways of selecting two more albums out of 11. So combining these we get 10*55=550 ways.
There are 5 ways of selecting 4 out 5 DK albums:
ABCD, ABCE, ABDE, ACDE, BCDE, making the total so far 555.
There are 10 ways of selecting 3 DK albums out of 5:
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. If we combine these with one album by a different artiste:
ABCa,...,ABCk,ABDa,...,...CDEk,
we get 10*11=110 ways, making 665.