1. **Problem Statement:** We have 32 students studying at least one of French (F), Spanish (S), or German (G). Given counts for each language and their intersections, we need to find $x$, the number of students studying both Spanish and German. 2. **Given Data:** - Total students $U = 32$ - $|F| = 20$ - $|S| = 18$ - $|G| = 12$ - $|F \cap S| = 10$ - $|F \cap G| = 6$ - $|S \cap G| = x$ - $|F \cap S \cap G| = 3$ - Every student studies at least one language. 3. **Important Formula:** The principle of inclusion-exclusion for three sets: $$|F \cup S \cup G| = |F| + |S| + |G| - |F \cap S| - |S \cap G| - |F \cap G| + |F \cap S \cap G|$$ Since every student studies at least one language, $|F \cup S \cup G| = 32$. 4. **Substitute known values:** $$32 = 20 + 18 + 12 - 10 - x - 6 + 3$$ 5. **Simplify the right side:** $$32 = 50 - 10 - x - 6 + 3$$ $$32 = 50 - 13 - x$$ $$32 = 37 - x$$ 6. **Solve for $x$:** $$x = 37 - 32 = 5$$ 7. **Interpretation:** The number of students studying both Spanish and German is $x = 5$. 8. **Verify with the Venn diagram values:** - Intersection $S \cap G = 5$ - Intersection $F \cap S = 10$ - Intersection $F \cap G = 6$ - Triple intersection $F \cap S \cap G = 3$ All values are consistent with the problem's handwritten notes. **Final answer:** $$\boxed{5}$$