In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?

$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 126 $
1. **Total Points Calculation**:
   - There are 10 runners in total (5 from each team).
   - The sum of the positions from 1 to 10 is given by the formula for the sum of the first \( n \) natural numbers:
     \[
     \sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55
     \]
   - Therefore, the total points distributed among all runners is 55.

2. **Score Distribution**:
   - Let \( S_A \) be the score of Team A and \( S_B \) be the score of Team B.
   - Since the total score is 55, we have:
     \[
     S_A + S_B = 55
     \]
   - The team with the lower score wins. Therefore, we need to find the possible values for \( S_A \) such that \( S_A < S_B \).

3. **Range of Scores**:
   - The minimum score for a team occurs when its runners finish in the first 5 positions:
     \[
     1 + 2 + 3 + 4 + 5 = 15
     \]
   - The maximum score for a team occurs when its runners finish in the last 5 positions:
     \[
     6 + 7 + 8 + 9 + 10 = 40
     \]
   - Therefore, the possible scores for a team range from 15 to 40.

4. **Winning Scores**:
   - For Team A to win, \( S_A \) must be less than \( S_B \). Given \( S_A + S_B = 55 \), we have:
     \[
     S_A < 27.5
     \]
   - Since \( S_A \) must be an integer, the highest possible winning score for Team A is 27.
   - The possible winning scores for Team A range from 15 to 27.

5. **Counting the Possible Winning Scores**:
   - The number of integers from 15 to 27 inclusive is:
     \[
     27 - 15 + 1 = 13
     \]

Conclusion:
\[
\boxed{13}
\]