7. Let $S=\{1,2,3, \ldots, 9,10\}$. A non-empty subset of $S$ is considered "Good" if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets $\{4,8\},\{3,4,7,8\}$ and $\{1,3,6,8,10\}$ are "Good". How many subsets of $S$ are "Good"?
(A) 482
(B) 507
(C) 575
(D) 637
(E) 667
7. Answer: (D)
Let the number of even integers in a "Good" subset of $S$ be $i$, where $i=1,2,3,4$, 5 , and the number of odd integers in that subset be $j$, where $j=0,1,2, \ldots, i$. Then the number of "Good" subsets of $S$ is
$$
\begin{aligned}
\sum_{i=1}^{5}\left(\binom{5}{i} \sum_{j=0}^{i}\binom{5}{j}=\right. & \binom{5}{1}\binom{5}{0}+\binom{5}{1}+\binom{5}{2}\binom{5}{0}+\binom{5}{1}+\binom{5}{2}+\cdots \\
& +\binom{5}{5}\binom{5}{0}+\binom{5}{1}+\cdots+\binom{5}{5} \\
& =5(1+5)+10(1+5+10)+\cdots+(1+5+10+10+5+1) \\
& =30+160+260+155+32=637 .
\end{aligned}
$$