Example 2 A class participated in a math competition, with a total of $a$, $b$, and $c$ three questions. Each question either scores full marks or 0 points, where question $a$ is worth 20 points, and questions $b$ and $c$ are worth 25 points each. After the competition, every student answered at least one question correctly, and those who answered all questions correctly are $\mathrm{i}$ people, and 15 people answered two questions correctly. The sum of the number of people who answered question $a$ correctly and the number of people who answered question $b$ correctly is 29; the sum of the number of people who answered question $a$ correctly and the number of people who answered question $c$ correctly is 25; the sum of the number of people who answered question $b$ correctly and the number of people who answered question $c$ correctly is 20. What is the average score of the class?
(1909, National Junior High School Mathematics Competition)
Solution: Let $x, y, z$ represent the number of correct answers for questions $a$, $b$, and $c$, respectively. Then we have
$$
\left\{\begin{array} { l } 
{ x + y = 2 9 , } \\
{ x + z = 2 5 , } \\
{ y + z = 2 0 . }
\end{array} \text { Solving, we get } \left\{\begin{array}{l}
x=17, \\
y=12, \\
z=8 .
\end{array}\right.\right.
$$

The number of people who answered 1 question correctly is
$$
37-1 \times 3-2 \times 15=4 \text { (people). }
$$

The total number of people in the class is $1+4+15=20$ (people).
Thus, the average score is
$$
\frac{17 \times 20+(12+8) \times 25}{20}=42 \text { (points). }
$$