# CD collection

How many different ways can I arrange the CDs in my collection?

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*

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.