1. **State the problem:** We have a class of 39 students. - 19 students offer French. - 25 students offer German. - Some students do not offer any language. We need to find: - How many students offer both languages. - How many students offer only French (not German). 2. **Define variables and use the principle of inclusion-exclusion:** Let: - $F$ = number of students offering French = 19 - $G$ = number of students offering German = 25 - $N$ = total number of students = 39 - $B$ = number of students offering both French and German - $O_F$ = number of students offering only French - $O_G$ = number of students offering only German - $N_0$ = number of students offering neither language 3. **Use the formula for total students:** $$N = O_F + O_G + B + N_0$$ 4. **Express $O_F$ and $O_G$ in terms of $B$:** $$O_F = F - B$$ $$O_G = G - B$$ 5. **Substitute into total:** $$N = (F - B) + (G - B) + B + N_0 = F + G - B + N_0$$ 6. **Rearrange to solve for $B$:** $$B = F + G + N_0 - N$$ 7. **Find $N_0$ (students offering neither):** Given in the problem: "first students do not offer any of a two language" is ambiguous, but assuming it means some students do not offer any language. Since $N=39$, $F=19$, $G=25$, and $N_0$ is unknown, but total students cannot be less than sum of those offering languages minus overlap. 8. **Calculate $B$ assuming $N_0=5$ (since $19 + 25 = 44$ which is 5 more than 39):** $$B = 19 + 25 - 39 = 44 - 39 = 5$$ 9. **Calculate only French students:** $$O_F = F - B = 19 - 5 = 14$$ 10. **Final answers:** - Students offering both languages: $5$ - Students offering only French: $14$