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Math Help - Counting

  1. #1
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    Counting

    10. In a 6-cylinder automobile engine, the even numbered cylinders are on the engine’s left side and the odd numbered cylinders are on the right side. (Drawing a picture will help you to visualize this.) A good firing order is a permutation of the numbers 1 to 6 in which the right and left sides are alternated.
    (a) How many possible good firing orders are there? (b) Repeat this problem for an engine with 2n cylinders.

    Any thoughts? Thanks.
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  2. #2
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    Hi cgiulz.

    Quote Originally Posted by cgiulz View Post
    10. In a 6-cylinder automobile engine, the even numbered cylinders are on the engine’s left side and the odd numbered cylinders are on the right side. (Drawing a picture will help you to visualize this.) A good firing order is a permutation of the numbers 1 to 6 in which the right and left sides are alternated.
    (a) How many possible good firing orders are there?
    There are 3 cylinders on the left side and 3 cylinders on the right side.

    firing a cylinder on the right side, there are 3 possibilities. After that you have to fire one on the left side. there are 3 possibilities, too.
    Now you are back on the right side, there are 2 possibilities (because one cylinder already is fired) - on the left side there are 2 possibilities.

    Therefore

    Probability("firing right, firing left, firing right, left, right left") = 3*3*2*2*1*1

    I guess you can start firing a cylinder on the left side, too, therefore

    Probability("firing left, right, left right left, right= 3*3*2*2*1*1

    and the solution therefore is

    2*(3*3*2*2*1*1)

    Quote Originally Posted by cgiulz View Post
    (b) Repeat this problem for an engine with 2n cylinders.

    Any thoughts? Thanks.
    Do you have any thoughts on (b) now?

    Rapha
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  3. #3
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    2*(n^2*(n-1)^2....1*1).

    Thanks, you've been super helpful .
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