5. Consider all non-empty subsets of the set $S=\{1,2, \cdots, 10\}$. If the number of even numbers in a non-empty subset is no less than the number of odd numbers, that subset is called a "good subset". The number of "good subsets" is [
(A) 631
(B) 633
(C) 635
(D) 637
Choose $D$.
Suppose a "good subset" contains $i(i=1,2,3,4,5)$ even numbers, then the number of odd numbers can be $j=0,1, \cdots, i$, so the number of "good subsets" is
$$
\begin{array}{l}
\sum_{i=1}^{5}\left(C_{5}^{i} \sum_{j=0}^{i} C_{5}^{j}\right)=C_{5}^{1}\left(C_{5}^{0}+C_{5}^{1}\right)+C_{5}^{2}\left(C_{5}^{0}+C_{5}^{1}+C_{5}^{2}\right)+C_{5}^{3}\left(C_{5}^{0}+C_{5}^{1}+C_{5}^{2}+C_{5}^{3}\right) \\
\quad+C_{5}^{4}\left(C_{5}^{0}+C_{5}^{1}+C_{5}^{2}+C_{5}^{3}+C_{5}^{4}\right)+C_{5}^{5}\left(C_{5}^{0}+C_{5}^{1}+C_{5}^{2}+C_{5}^{3}+C_{5}^{4}+C_{5}^{5}\right)=637
\end{array}
$$