1. **Problem statement:** We have a survey of 100 people with the following data: - Joggers (J) = 50 - Swimmers (S) = 30 - Cyclists (C) = 35 - Jog and Swim (J \cap S) = 14 - Swim and Cycle (S \cap C) = 7 - Jog and Cycle (J \cap C) = 9 - All three (J \cap S \cap C) = 3 We want to find: a. Number who jog but do not swim or cycle. b. Number who take part in only one activity. c. Number who do not take part in any activity. 2. **Formula and rules:** - Use the principle of inclusion-exclusion for three sets: $$|J \cup S \cup C| = |J| + |S| + |C| - |J \cap S| - |S \cap C| - |J \cap C| + |J \cap S \cap C|$$ - To find only one activity participants, subtract those in intersections. 3. **Calculate number who jog only:** $$|J \text{ only}| = |J| - |J \cap S| - |J \cap C| + |J \cap S \cap C|$$ Substitute values: $$= 50 - 14 - 9 + 3 = 30$$ 4. **Calculate number who swim only:** $$|S \text{ only}| = |S| - |J \cap S| - |S \cap C| + |J \cap S \cap C| = 30 - 14 - 7 + 3 = 12$$ 5. **Calculate number who cycle only:** $$|C \text{ only}| = |C| - |J \cap C| - |S \cap C| + |J \cap S \cap C| = 35 - 9 - 7 + 3 = 22$$ 6. **Number who take part in only one activity:** $$= |J \text{ only}| + |S \text{ only}| + |C \text{ only}| = 30 + 12 + 22 = 64$$ 7. **Calculate total number who take part in at least one activity:** $$|J \cup S \cup C| = 50 + 30 + 35 - 14 - 7 - 9 + 3 = 88$$ 8. **Number who do not take part in any activity:** $$= 100 - |J \cup S \cup C| = 100 - 88 = 12$$ **Final answers:** a. 30 people jog but do not swim or cycle. b. 64 people take part in only one activity. c. 12 people do not take part in any of these activities.