Problem solving a word problem using a generating function 
How many ways are there to hand out 24 cookies to 3 children so that they each get an even number, and they each get at least 2 and no more than 10? Use generating functions. 

So the first couple steps are easy.
The coefficient is $x^{24}$
$g(x) = x^6(1+x^2+x^4+x^6+x^8)^3$ or  what I got was $x^6 (1 + (x^2)^1 +...+ (x^2)^4)^3$
now finding the closed formula is where I am having problems 
My answer: using the fact that $\dfrac{1-x^{n+1}}{1- x}$
I get $x^6\left(\dfrac{1-x^9}{1-x}\right)$ which is wrong 
The correct answer: $x^6\left(\dfrac{1-x^{10}}{1-x^2}\right)$
If someone could explain in some detail on how to get the correct formula would be much appreciated. Thanks!

Here is variation of the theme which might be helpful when doing the calculation. It's convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ of a series.

We obtain
  \begin{align*}
[x^{24}]&(x^2+x^4+x^6+x^8+x^{10})^3\tag{1}\\
&=[x^{24}]x^6(1+x^2+x^4+x^6+x^8)^3\\
&=[x^{18}]\left(\frac{1-x^{10}}{1-x^2}\right)^3\tag{2}\\
&=[x^{18}](1-x^{10})^{3}\sum_{n=0}^\infty\binom{-3}{n}(-x^2)^n\tag{3}\\
&=[x^{18}](1-3x^{10})\sum_{n=0}^\infty\binom{n+2}{2}x^{2n}\tag{4}\\
&=\left([x^{18}]-3[x^8]\right)\sum_{n=0}^\infty\binom{n+2}{2}x^{2n}\tag{5}\\
&=\binom{11}{2}-3\binom{6}{2}\tag{6}\\
&=10
\end{align*}

Comment:


*

*In (1) we respect that an even number of at least $2$ and at most $10$ cookies is to distribute to three children. The coefficient of operator selects the coefficient of $x^{24}$ which gives the wanted number.

*In (2) we use the rule
\begin{align*}
[x^{p+q}]A(x)=[x^p]x^{-q}A(x)
\end{align*}
and apply the finite geometric series with argument $x^2$.

*In (3) we use the binomial series expansion for $\frac{1}{(1-x^2)^3}$

*In (4) we expand $(1-x^{10})^3$ and take the first two terms only since all other therms have powers greater than $18$. We also use the binomial identity
\begin{align*}
\binom{-p}{q}=\binom{p+q-1}{p-1}(-1)^q
\end{align*}

*In (5) we use the linearity of the coefficient of operator and apply the rule stated in the comment of (2).

*In (6) we select the coefficients of $x^{18}$ and $x^{8}$ by taking $n=9$ and $n=4$.
