From a set of nine colored balls, including three red, four blue, and two green, five balls are selected. However, it is required that at least one red ball is included and that the number of blue balls included is even. Use a generating function to find how many ways this can be done.
The given g.f expression is $(R + R^2 + R^3)(1 + B^2 + B^4)(1 + G + G^2)$ which is actually $x(1 + x + x^2)^2(1 + x^2 + x^4)$.
I thought the g.f. should be $(R + R^2 + R^3)(1 + B^2 + B^4)(1 + G + G^2 + G^3)$. For example, we can choose $2$ red, $3$ green and $0$ blue balls because $0$ is even.
Can someone, please, explain why the third factor in the product shouldn't have the term $G^3?$ Thanks.
The number of green balls in the bag is two and hence the maximum number of green balls we can select is two.