1. **Problem Statement:** We have a survey about people liking apricots (A), bananas (B), and cantaloupes (C) with given counts and intersections. We want to find the number who liked all three fruits ($x$), total interviewed, and those who liked bananas and apricots but not cantaloupes. 2. **Given Data:** - $|A|=39$ - $|B|=50$ - $|C|=39$ - $|A \cap B|=21$ - $|B \cap C|=18$ - $|A \cap C|=19$ - Number who liked exactly two fruits = 22 - Number who liked none = 8 3. **Key Formulas and Rules:** - Number who liked exactly two fruits is sum of pairwise intersections minus thrice the triple intersection: $$|A \cap B| + |B \cap C| + |A \cap C| - 3x = 22$$ - Total number interviewed is union plus those who liked none: $$|A \cup B \cup C| + 8$$ - Inclusion-Exclusion Principle for union: $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + x$$ 4. **Find $x$:** From exactly two fruits: $$21 + 18 + 19 - 3x = 22$$ $$58 - 3x = 22$$ $$3x = 58 - 22 = 36$$ $$x = \frac{36}{3} = 12$$ 5. **Find total number interviewed:** Calculate union: $$|A \cup B \cup C| = 39 + 50 + 39 - 21 - 18 - 19 + 12 = 128 - 58 + 12 = 82$$ Add those who liked none: $$82 + 8 = 90$$ 6. **Find number who liked bananas and apricots but NOT cantaloupes:** This is the part of $|A \cap B|$ excluding those who liked all three: $$|A \cap B| - x = 21 - 12 = 9$$ **Final answers:** - $x = 12$ - Total interviewed = 90 - Bananas and apricots but not cantaloupes = 9