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Homework Help: Composition problem help

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data

    An astronaut crew is preparing for a space mission. As part of their preparation,
    they must spend a block of consecutive days training in a flight simulator. Each day,
    the crew is required to spend either 1, 2, or 3 hours in the simulator. The training
    ends once they attain n hours of simulator time.
    How many ways can the crew achieve the required n hours of training? Express your
    answer as a coefficient of a generating function.

    2. Relevant equations
    Generating functions. sum lemma. product lemma.

    3. The attempt at a solution

    Let Sk be the set Sk={1,2,3}^k
    and let S= U(union) Sk

    By the sum and product lemmas
    Is(x) (the generating function)= sigma Isk(x)
    k>= 0
    = sigma (I (x))^k
    k>=0 {1,2,3}
    = sigma (x+x^2+x^3)^k

    = 1/ (1- (x+x^2+x^3))

    therefore the number of ways the crew can achieve the required n hours of training is

    have i made any mistakes. am i completely off? any help would be greatly appreciated.
  2. jcsd
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