Solving the Composition Problem for Astronaut Training with Generating Functions

[beddytear]

Homework Statement

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.

Homework Equations

Generating functions. sum lemma. product lemma.

The Attempt at a Solution

Let Sk be the set Sk={1,2,3}^k
and let S= U(union) Sk
k>=0

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
k>=0

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

therefore the number of ways the crew can achieve the required n hours of training is
[x^n](coefficient)(1/(1-(x+x^2+x^3)).

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

 

[mighty2000]

Your solution is correct! You have used the correct generating function and the sum and product lemmas to find the coefficient of x^n, which represents the number of ways the crew can achieve the required n hours of training. Good job!