<s>[INST] Please help to find term's coefficient in the following example I trying find the number of all solutions in the following:
$ x_1 + x_2 + x_3 + x_4 + x_5 = 24 $
where: 
2 of variables are natural odd numbers
3 of variables are natural even numbers
none of variables are equal to $0$ or $1$
(all the variables are $>= 2$)
So I've made generating functions according to the limitations as follow:
$ (x^3 + x^5 + x^7 + ... + x^{21} + x^{23})^2 $
$ (x^2 + x^4 + x^6 + ... + x^{22} + x^{24})^3 $
Now I need to find the coefficient of $ x^{24} $
I've tried to solve it but get stuck, here is what I've done:
$ x^6(1 + x^2 + x^4 + ... + x^{18} + x^{20})^2 $
$ x^6(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $
$ x^{12}(1 + x^2 + x^4 + ... + x^{18} + x^{20})^2(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $
Now we actually looking for coefficient of $ x^{24-12}=x^{12} $ in:
$ (1 + x^2 + x^4 + ... + x^{18} + x^{20})^2(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $
$ (\frac{1 - x^{21}}{1 - x})^2(\frac{1 - x^{23}}{1 - x})^3 $
Here I'm stuck. How to proceed in order to find coefficient of $ x^{12} $?
Regards.
PS: Solution was given but just out of curiosity, how to solve it using generating-functions?
 [/INST] Let the numbers be $2a + 1, 2b + 1, 2c, 2d, 2e$, where $a, b, c, d, e$ are natural. Note that this makes the odd numbers greater than or equal to $3$, and the even numbers greater than or equal to $2$.
Then you have $2a + 1 + 2b + 1 + 2c + 2d + 2e = 24 \rightarrow a + b + c + d + e = 11$. You can solve this with a standard stars-and-bars approach.
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