Example 2-35 A unit has 8 male colleagues and 5 female colleagues. It is planned to organize a group with an even number of male colleagues and no less than 2 female colleagues. How many different ways are there to form the group?
\[
\binom{8}{2}=\binom{8}{0}=28,\binom{8}{4}=70, \text{ let the generating function for male colleagues be}
\]
\[
A(x)=1+28 x^{2}+70 x^{4}+28 x^{6}+x^{8}
\]

Similarly, the generating function for the allowed combinations of female colleagues is
\[
\begin{aligned}
B(x)= & 10 x^{2}+10 x^{3}+5 x^{4}+x^{5} \\
C(x)= & A(x) B(x)=\left(1+28 x^{2}+70 x^{4}+28 x^{6}+x^{8}\right) \\
& \times\left(10 x^{2}+10 x^{3}+5 x^{4}+x^{5}\right) \\
= & 10 x^{2}+10 x^{3}+285 x^{4}+281 x^{5}+840 x^{6}+728 x^{7} \\
& +630 x^{8}+350 x^{9}+150 x^{10}+38 x^{11}+5 x^{12}+x^{13}
\end{aligned}
\]
The coefficient \( c_{k} \) of the \( x^{k} \) term in \( C(x) \) is the number of valid groups of \( k \) people, and the total number of group formation schemes is
\[
10+10+285+281+840+728+630+350+150+38+5+1=3328 .
\]