17. The numbers $1,2,3, \cdots, 7$ are randomly divided into two non-empty subsets. The probability that the sum of the numbers in the two subsets being equal is $\frac{p}{q}$ expressed in the lowest term. Find $p+q$.
17. Answer: 67
The total number of ways of dividing the seven numbers into two non-empty subsets is $\frac{2^{7}-2}{2}=63$. Note that since $1+2+3+\cdots+7=28$, the sum of the numbers in each of the two groups is 14 . Note also that the numbers $5,6,7$ cannot be in the same group since $5+6+7=18>14$. We consider three separate cases:
Case (i): Only 6 and 7 in the same group and 5 in the other group:
$$
\{2,3,4,5\},\{1,6,7\}
$$

Case (ii): Only 5 and 6 in the same group and 7 in the other group:
$$
\begin{array}{l}
\{1,2,5,6\},\{3,4,7\} \\
\{3,5,6\},\{1,2,4,7\}
\end{array}
$$

Case (iii): Only 5 and 7 in the same group and 6 in the other group:
$$
\{2,5,7\},\{1,3,4,6\}
$$

Hence there are 4 such possibilities. Thus the required probability is $\frac{4}{63}$, yielding that $p+q=67$.