10. (20 points) In the first month, there is a pair of adult rabbits. Every month, they breed 3 male rabbits and 2 female rabbits. The young female rabbits grow up and also breed 3 male rabbits and 2 female rabbits every other month. Question: How many rabbits will there be in the $n$ $\left(n \in \mathbf{Z}_{+}\right)$-th month?
10. Let $a_{n}$ denote the number of male rabbits in the $n$-th month, and $b_{n}$ denote the number of female rabbits in the $n$-th month. Then, from the problem statement, we have
$$
\left\{\begin{array}{l}
a_{n}=a_{n-1}+3 b_{n-2}, \\
b_{n}=b_{n-1}+2 b_{n-2},
\end{array}\right.
$$

where $n \geqslant 2, a_{0}=1, a_{1}=4, b_{0}=1, b_{1}=3$.
Solving this, we get $a_{n}=2^{n+1}-\frac{1-(-1)^{n-1}}{2}$,
$$
b_{n}=\frac{2^{n+2}-(-1)^{n}}{3} \text {. }
$$

Then $a_{n}+b_{n}$
$$
\begin{array}{l}
=2^{n+1}-\frac{1-(-1)^{n-1}}{2}+\frac{2^{n+2}-(-1)^{n}}{3} \\
=\frac{5 \times 2^{n+2}-5(-1)^{n}-3}{6} .
\end{array}
$$

Therefore, after the $n$-th month, there are $\frac{5 \times 2^{n+2}-5(-1)^{n}-3}{6}$ rabbits.