1. In the room, there are several people who know at least one of the three languages: English, French, and German. One-sixth of them know English, one-sixth know German, one-seventh know French, four know English and German, three know German and French, two know French and English, and one knows all three languages. How many people are in the room? How many of them know only English?
Solution. The problem is most simply solved using a Venn diagram. If we denote by $A, \Phi$, and $\Gamma$ the sets of students who speak English, French, and German, respectively, then from the conditions of the problem, we easily obtain the diagram shown in the figure on the right. According to this, in the room there were

![](https://cdn.mathpix.com/cropped/2024_06_05_d149e2e33a82a010b78cg-1.jpg?height=274&width=291&top_left_y=764&top_left_x=1191)

$$
1+1+3+2+1+3+0=11
$$

people who know at least one language, and since there were no other people in the room, we conclude that there were a total of 11 people in the room, and only one person knows only English.