1. **Problem statement:** Given sets with the following values: $|A|=55$, $|B|=40$, $|C|=80$, $|A \cap B|=20$, $|A \cap B \cap C|=17$, $|B \cap C|=24$, and $|A \cup C|=100$, find: 1. $|A \cap C|$ 2. $|C - B|$ 3. $|B \cap C| - |A \cap B \cap C|$ 2. **Formulas and rules:** - The union of two sets: $$|A \cup C| = |A| + |C| - |A \cap C|$$ - The difference of sets: $$|C - B| = |C| - |B \cap C|$$ - Intersection subtraction is straightforward. 3. **Step-by-step solution:** **Step 1: Find $|A \cap C|$ using the union formula:** $$|A \cup C| = |A| + |C| - |A \cap C|$$ Plug in the values: $$100 = 55 + 80 - |A \cap C|$$ Simplify: $$100 = 135 - |A \cap C|$$ Rearranged: $$|A \cap C| = 135 - 100 = 35$$ **Step 2: Find $|C - B|$ (elements in $C$ not in $B$):** $$|C - B| = |C| - |B \cap C| = 80 - 24 = 56$$ **Step 3: Calculate $|B \cap C| - |A \cap B \cap C|$:** $$24 - 17 = 7$$ 4. **Final answers:** 1. $|A \cap C| = 35$ 2. $|C - B| = 56$ 3. $|B \cap C| - |A \cap B \cap C| = 7$