Find the generating function for the number of ways to distribute blank scratch paper to Alice, Bob, Carlos, and Dave so that Alice gets at least two sheets, Bob gets at most three sheets, the number of sheets Carlos receives is a multiple of three, and Dave gets at least one sheet but no more than six sheets of scratch paper. Without finding the power series expansion for this generating function, determine the coefficients on x2 and x3 in this generating function.
I have no idea how to approach this...
Take each person as a separate variable and multiply together the combinations:
Alice = $x^2+x^3+\dots $
Bob = $1+x+x^2+x^3$
Carlos = $1+x^3+x^6+\dots$
Dave = $x+x^2+x^3+x^4+x^5+x^6$
The product of these could be a generating function that may be one idea to explore though I'd wonder if you need separate variables to know who got how many sheets or not.
Note: The minimum term in the product will be $x^3$ as Alice gets 2 sheets and Dave gets 1 sheet is the minimal solution, so I'd wonder what if the x2 is $x_2$ or $x^2$ as there are a couple of ways to see where a 2 could go here.