Thread: Number of permutations when composing recital

The problem is:

A pianist wants to prepare a recital that will consist of 3 Classical, 4 Romantic and 2 Contemporary pieces.
In how many ways can he order the piano pieces if he wants to play all the pieces of a genre together?

I'm a bit confused with these kind of problems, when the first choice could influence a second choice. If the first choice would be a Classical piece, then the following pieces must be Classical as well. Anyway, my reasoning was as follows, using the fundamental counting principle:

In total there are 9 "events (E's)", because there are 9 pieces of music in total.

E1 = 9 choices
E2 = 2 choices, if for example, the first choice (E1) was a Classical piece
E3 = then must be another Classical piece which is 1 remaining choice
E4 = 6 choices (4R + 2C)
E5 = if Romantic was chosen at E4, then here are 3 choices
E6 = 2 choices
E7 = 1 choice
E8 = 2 choices, only Contemporary left
E9 = 1 choice

So the total number of ways this can happen is: 9*2*1*6*3*2*1*2*1 = 1260 ways.

According to the answer sheet this is incorrect. The correct answer should be 1728 ways.

Can somebody show me where my reasoning if faulty?

Magically after writing my problem I think I found the right reasoning.

You can order the Classical pieces in 3*2*1 = 6 ways
The Romantic pieces in 4*3*2*1 = 24 ways
The Contemporary pieces in 2*1 = 2 ways

You can order the genres in 3*2*1 = 6 ways

Total number of ways = 6*24*2*6 = 1728 ways

Right!? :P

Right!