CD collection
How many different ways can I arrange the CDs in my collection?
Problem
I have 3 Elvis Presley CDs, 2 Beatles CDs and 1 Queen CD.
How many different ways can they be lined up on a shelf if I must keep all the CDs by the same artist together?
This problem is adapted from the World Mathematics Championships
Student Solutions
First, we need to decide which artist will go first, which will go second, and which will go third.
There are 3 options for the artist that goes first, and then 2 options remaining for the artist that goes second, and then only 1 option left for the artist that goes third. So there are 3$\times$2$\times$1 = 6 ways to decide which artist will go first, which will go second, and which will go third.
There are 2 Beatles CDs, so for each position that they could be in relative to the other CDs, there are 2 options for the Beatles CDs - because they could go either way round.
Similarly, since there are 3 Elvis Presley CDs, for each position that they could be in relative to the other CDs, there are 3 options for the first Elvis Presley CD, 2 options for the second and only 1 option for the third. So there are 3$\times$2$\times$1 = 6 ways to arrange the Elvis Presley CDs.
So altogether, there are 6$\times$2$\times$6 = 72 ways to arrange the CDs.