3. In a cross-country race, there are two teams participating, each with 5 members. When the athletes reach the finish line, their ranks are recorded, and the team of the $n$-th arriving athlete scores $n$ points, with the team having the lower total score winning. Now, assuming no two team members arrive at the finish line simultaneously, how many possible scores can the winning team have?
(40th American High School Mathematics Examination)
(The hint: The sum of the scores of 10 athletes is $1+2+\cdots+10$ $=55$, thus the total score of the winning team does not exceed $\left[\frac{55}{2}\right]=27$. Also, the total score of the winning team is no less than $1+2+\cdots+5=15$, it is easy to see that the score of the winning team can take all values from $15,16, \cdots, 27$. Therefore, the winning team's score has 27 $-15+1=13$ possible values.)