Find number of ways to choose $3n$-subset with repetitions from set $\left\{A,B,C\right\}$ such that:
1. Letter $A$ occur at most $2n$
2. Letter $B$ occur at most $2n$
3. Letter $C$ occur odd times
Approach
I want to use there enumerators. Ok, so a factor responsible for $A$ will be
$$(1+x+x^2+ \cdots + x^{2n}) $$
(We can choose $A$ $0$ times, $1$ time, ... $2n$ times). The same will be for $B$.
Enumerator for $C$ will be
$$(x+x^3+x^5 + \cdots) $$
(We can choose $C$ 1 time, 3 times, etc)
Ok, so I want to find
$$[x^{3n}](1+x+x^2+ \cdots + x^{2n})(1+x+x^2+ \cdots + x^{2n})(x+x^3+x^5 + \cdots) = $$
$$ [x^{3n}] \left(\frac{1-x^{2n+1}}{1-x}\right)^2 \cdot\frac{x}{1-x^2} $$
but... how I can get from there factor at $x^{3n}$?