How many ways are there to fill a bag (maximum of weight $N$) with watermelons, apples, and grapes, where the number of apples has to be at least the number of watermelons? The weights are given as follows:
$\omega(watermelon) = 10$
$\omega(apple) = 2$
$\omega(grapes) = 1$
So far, I've tried creating the series for each fruit and using the product lemma to multiply them. The answer would be:
the coefficient of $x^N$ of $(\sum_{j=0}^{\inf} x^j)\ (\sum_{j=0}^{inf} x^{2j})\ (\sum_{j=0}^{inf} x^{10j})$
However, this does not account for the restriction that the number of apples has to be at least the number of watermelons. I don't really know where to go from here.
Edit: Thank you Nicholas for the answer. I found another way to get the same answer. I'd say this is more of an algebraic solution where as Nicholas' solution is more combinatoric. Here is my approach: let a = #apples, w = #watermelons, g = #grapes
we know that $a = w +y$, where $y \in Z, y\ge 0$. and $g + 2a + 10w = N$, substituting we get $g + 2y + 12w$ = N.
We can now write this in terms of generating functions as $(\sum_{j=0}^{\inf} x^j)\ (\sum_{j=0}^{inf} x^{2j})\ (\sum_{j=0}^{inf} x^{12j})$, which gives the same solution.
Because of the closely connected nature of the numbers of apples and watermelons, you can't handle them separately. I would say that the OGF for the total weight of apples AND watermelons is $$ \sum_{n=0}^{\infty}x^{10n}\sum_{m=n}^{\infty}x^{2m}=\sum_{n=0}^{\infty}x^{10n}\cdot\frac{x^{2n}}{1-x^2}=\frac{1}{1-x^2}\sum_{n=0}^{\infty}x^{12n}=\frac{1}{(1-x^2)(1-x^{12})}. $$
Another way to explain this generating function: you are taking any number of pairs of apples and watermelons, at a total of weight 12 each, plus any number of apples at weight 2.
In any event, since the number of grapes IS separate from apples/watermelons, you then just have to multiply by the OGF for grapes, which is $\frac{1}{1-x}$.