Problem 3. The students from $\mathrm{IV}^{\mathrm{a}}$ class are members of the environmental, literary, or math section. Five students are members of all three sections, and nine students are members of two sections. Eight students are members of both the environmental and literary sections, and the same number are members of both the literary and math sections. Also, 20 students are members of only one section, with 5 in the environmental and math sections. How many students are there in IV ${ }^{\mathrm{a}}$ class?
Solution. First method. Using Euler-Venn diagram

![](https://cdn.mathpix.com/cropped/2024_06_05_9d1696c7fd1a2cbf5bb5g-04.jpg?height=486&width=1007&top_left_y=668&top_left_x=415)

We have, $\left|I V^{a}\right|=5+3+3+3+1+5+10=34$.

Second method. In all three sections, 5 students are enrolled. Since in every two sections, 8 students are enrolled, then only in two sections (excluding students who are enrolled in all three sections simultaneously), 3 students are enrolled. In the school section, a total of $5+3+3+5=16$ students are enrolled. Similarly, 16 students are enrolled in the mathematics section. According to the condition of the problem, only in the literature section, $20-5-5=10$ students are enrolled, and in total, 21 students are enrolled in that section. In the class, there are $5+3+3+3+10+5+5=34$ students.