A basket holds 10 bananas, 8 apples, 6 oranges and 4 pears. Let $a_i$ be the number of different presents composed of $i$ fruits. Each such present contains at least 1 banana, at most 2 apples, the number of oranges in a present is even, and the number of pears in a present is odd. Create a generating function for ${\{{a_i\}}^\infty_{i=1}}$.
I'm not really sure how to approach this problem, I'd appreciate any guidance.
The generating function \begin{eqnarray*} (b+b^2+\cdots+b^{10})(1+a+a^2)(1+o^2+o^4+o^6)(p+p^3) \end{eqnarray*} will give the number of choices of fruits and keep tabs on which type of each fruit.
We are only intrested in the number of configurations for a given number of fruits, so let $a=b=o=p=x$ and we have \begin{eqnarray*} a_i= [x^i]:(x+x^2+\cdots+x^{10})(1+x+x^2)(1+x^2+x^4+x^6)(x+x^3). \end{eqnarray*}