Generating functions question: There are n objects - rings, earring and bracelets. How many ways are there to arrange these objects, as the amount of earring is even and there are at most 4 bracelets.
[This is a question we must solve with generating functions].
Here's what I started:
There are $n$ objects, at most 4 bracelets. There are $2k$ bracelets when $2k \leq n$. So there are $n-2k-4$ rings at most.
I defined the following generating function: $$A(x)=(x^0+x^1+...+x^{n-2k-4})(x^0+x^2+...+x^{2k})(x^0+x^1+x^2+x^3+x^4)={1-x^{n-2k-3}\over 1-x}\cdot {1-x^{2k+1}\over 1-x^2}\cdot {1-x^5 \over 1-x}$$
So I tried simplfying it... and.... well it didn't go well.
Is there another way to solve this? Is my direction even correct?
Thanks for any input!
You shouldn’t be building $n$ into your generating function: you actually want the coefficient of $x^n$ in the (correct) generating function, which is
$$A(x)=\left(\sum_{k\ge 0}x^k\right)\left(\sum_{k\ge 0}x^{2k}\right)\left(x^0+x^1+x^2+x^3+x^4\right)\;.$$
The first factor is for rings, of which you may have any number, the second is for earrings, of which you must have an even number, and the last is for bracelets, of which you must have at most $4$. Thus,
$$A(x)=\frac1{1-x}\cdot\frac1{1-x^2}\cdot\frac{1-x^5}{1-x}=\frac{1-x^5}{(1-x)^3(1+x)}\;.$$
Divide out, expand the remaining rational function into partial fractions, convert to power series, and find the coefficient of $x^n$.