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Math Help - Number of permutations when composing recital

  1. #1
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    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?
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  2. #2
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    Re: Number of permutations when composing recital

    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
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  3. #3
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    Re: Number of permutations when composing recital


    Right!
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