Say we have red, green, and blue marbles that we are arranging in a line of length n. We need to use an even number of blue marbles, at least two red marbles, and at most two green marbles. I am trying to count the number of total colourings using an exponential generation function: $\phi(t)=\sum_{n\ge0}\frac{f_nt^n}{n!}$ where $f_n$ is the number of total colourings.
The problem that I am running into with this question is that I also need to arrange the marbles once I select them. I believe I can select them with the egf:
$(\sum_{n\ge0}\frac{t^{2n}}{2n!})(\sum_{n\ge2}\frac{t^{n}}{n!})(1/0!+t/1!+t^2/2!)=\frac{e^t+e^{-t}}{2}*(e^t-1-t)*(1+t+t^2/2)$
Am I on the wrong track?