$36 \cdot 62$ In a cross-country competition, each side has 5 members. If a member finishes in the $n$th place, they score $n$ points for their team, and the team with fewer points wins. If there is no relationship between the members, then the number of possible winning scores for a team is
(A) 10.
(B) 13.
(C) 27.
(D) 120.
(E) 126.
(40th American High School Mathematics Examination, 1989)
[Solution] The total score after the two teams compete is
$$
1+2+3+\cdots+9+10=\frac{10 \cdot 11}{2}=55,
$$

thus the score of the winning team should not exceed $\left[\frac{55}{2}\right]=27$.
Furthermore, the minimum score for each team is $1+2+3+4+5=15$, so the possible scores for the winning team are between 15 and 27, i.e., $15,16, \cdots, 26,27$, a total of 13.
Therefore, the answer is $(B)$.