1. **State the problem:** We have 120 students surveyed about their preferences for three brands of mineral water: Voltaic, Silver, and Deep. Given the numbers who like each brand and their intersections, we want to find how many students like none of the three brands. 2. **Given data:** - Total students, $N = 120$ - Like Voltaic, $|V| = 52$ - Like Silver, $|S| = 58$ - Like Deep, $|D| = 58$ - Like both Voltaic and Silver, $|V \cap S| = 25$ - Like both Silver and Deep, $|S \cap D| = 18$ - Like both Voltaic and Deep, $|V \cap D| = 21$ - Like all three, $|V \cap S \cap D| = 13$ 3. **Formula used:** To find the number of students who like none, we use the principle of inclusion-exclusion: $$ |V \cup S \cup D| = |V| + |S| + |D| - |V \cap S| - |S \cap D| - |V \cap D| + |V \cap S \cap D| $$ 4. **Calculate the union:** $$ |V \cup S \cup D| = 52 + 58 + 58 - 25 - 18 - 21 + 13 $$ $$ = 168 - 64 + 13 = 117 $$ 5. **Calculate students who like none:** $$ \text{None} = N - |V \cup S \cup D| = 120 - 117 = 3 $$ **Final answer:** 3 students like none of the three brands. Note: Part (a) asks for a Venn diagram illustration which is not possible here, but the numbers above can be used to draw it.