1. **Problem statement:** We have 40 teens who watched Harry Potter. - 22 saw it on DVD - 23 saw it on video - 17 saw it in cinema - 7 saw it on DVD and cinema - 12 saw it on DVD and video - 6 saw it on video and cinema We want to find how many saw it on DVD, cinema, and video all three. 2. **Formula used:** Use the principle of inclusion-exclusion for three sets $A$, $B$, and $C$: $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$ 3. **Assign sets:** - $A$ = DVD watchers = 22 - $B$ = Video watchers = 23 - $C$ = Cinema watchers = 17 - $|A \cap B|$ = DVD and video = 12 - $|A \cap C|$ = DVD and cinema = 7 - $|B \cap C|$ = Video and cinema = 6 - $|A \cup B \cup C|$ = total who watched = 40 4. **Plug values into formula and solve for $|A \cap B \cap C|$:** $$40 = 22 + 23 + 17 - 12 - 7 - 6 + |A \cap B \cap C|$$ $$40 = 62 - 25 + |A \cap B \cap C|$$ $$40 = 37 + |A \cap B \cap C|$$ 5. **Isolate $|A \cap B \cap C|$:** $$|A \cap B \cap C| = 40 - 37 = 3$$ **Answer:** 3 teens have seen the movie on DVD, cinema, and video all three.