1. **Stating the problem:** We have three sets representing housewives using detergents A, B, and C, with given numbers and intersections. We want to find: a) The number of housewives who used all three brands. b) The number who used exactly two brands. c) The number who used three brands only (same as a). 2. **Given data:** \( |A|=42, |B|=50, |C|=48 \) \( |A \cap B|=12, |B \cap C|=18, |A \cap C|=13 \) 3. Let \( x = |A \cap B \cap C| \) be the number of housewives using all three brands. 4. Using the Inclusion-Exclusion principle for three sets: $$ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C| $$ Since total housewives surveyed = 100, assuming all use at least one brand, $$ 100 = 42 + 50 + 48 - 12 -18 -13 + x $$ Calculate: $$ 100 = 140 - 43 + x = 97 + x $$ So, $$ x = 100 - 97 = 3 $$ 5. Number using all three brands is \( \boxed{3} \). 6. Number using exactly two brands is each pair intersection minus those using all three: \( |A \cap B| - x = 12 - 3 = 9 \) \( |B \cap C| - x = 18 - 3 = 15 \) \( |A \cap C| - x = 13 - 3 = 10 \) Summing these: $$ 9 + 15 + 10 = 34 $$ 7. Number using three brands only is the same as all three brands: \( 3 \). **Final answers:** a) 3 b) 34 c) 3