There are 5 rock songs, 6 romantic songs, and 3 musicals.

This is a nice problem to test your concepts. Only for the high scorers.
First thing, an album can contain just one song. In this case, there is no definite set of songs in an album.

Number of ways you can pick rock songs out of the set of 5 * Number of ways you can pick romantic songs out of the set of 6 * Number of ways you can pick musicals out of the set of 3 = total number of possible albums

So, how many different ways can we pick rock songs out of the set of five.
0 songs can be picked in 5C0 ways = 1
1 songs can be picked in 5C1 ways= 5
2 songs can be picked in 5C2 ways= 10
3 songs can be picked in 5C3 ways = 10
4 songs can be picked in 5C4 ways = 5
5 songs can be picked in 5C5 ways = 1

Total number of ways = 1+5+10+10+5+1. Now, remember that we should have atleast one rock song, so our total number of ways is 31

Similarly, 6 romantic songs can be picked in 6C0 + 6C1 + 6C2 + 6C3 +6C4 + 6C5 + 6C6 = 1 + 6+ 15 + 20 + 6+ 1 = 49 , but we have to keep atleast 1 romantic song, so our total number of ways is 48

Similarly, 3 musicals can be picked in 3C0+3C1+3C2+3C3 ways = 1+ 3+3+1 = 8 ways

Then, the total number of albums is just the multiplication of the individual possibilities, 31 * 48 * 8 = 15624

Praetorian

Agree with Praet's method.

The basic answer pattern to this kind of question is the product of combs sums.

here :
(5C1+5C2+5C3+5C4+5C5).(6C1+6C2+6C3+6C4+6C5+6C6).(3C0+3C1+3C2+3C3)
=63*31*8=15624

A it is.