20. In a colfege mathematics class, all the students are also taking anthropology, history, or psychofogy, and some of the stu- dents are taking two, or even all three, of these courses, If (i) 40 students are taking anthropology, (ii) 11 students are taking history, (iii) 12 students are taking psychology, (iv) 3 students are taking all three courses, (v) 6 students are taking anthro- pology and history, and (vi) 6 students are taking psychology and anthropology, (a) how many students are taking only anthropology? (b) how many students are taking anthropology or history? (c) how many students are taking history and anthropology but not psychology?

To solve this problem, we can use the principle of inclusion-exclusion and set notation. Let's define the following sets: - Let \( A \) be the set of students taking Anthropology. - Let \( H \) be the set of students taking History. - Let \( P \) be the set of students taking Psychology. From the problem, we have the following information: 1. \( |A| = 40 \) (students taking Anthropology) 2. \( |H| = 11 \) (students taking History) 3. \( |P| = 12 \) (students taking Psychology) 4. \( |A \cap H \cap P| = 3 \) (students taking all three courses) 5. \( |A \cap H| = 6 \) (students taking both Anthropology and History) 6. \( |A \cap P| = 6 \) (students taking both Anthropology and Psychology) We can use this information to find the answers to the questions. ### (a) How many students are taking only Anthropology? To find the number of students taking only Anthropology, we can use the formula: \[ |A \text{ only}| = |A| - |A \cap H| - |A \cap P| + |A \cap H \cap P| \] Substituting the values we have: \[ |A \text{ only}| = 40 - 6 - 6 + 3 = 31 \] So, **31 students are taking only Anthropology.** ### (b) How many students are taking Anthropology or History? To find the number of students taking either Anthropology or History, we can use the inclusion-exclusion principle: \[ |A \cup H| = |A| + |H| - |A \cap H| \] Substituting the values we have: \[ |A \cup H| = 40 + 11 - 6 = 45 \] So, **45 students are taking Anthropology or History.** ### (c) How many students are taking History and Anthropology but not Psychology? To find the number of students taking both Anthropology and History but not Psychology, we can use the formula: \[ |A \cap H \text{ only}| = |A \cap H| - |A \cap H \cap P| \] Substituting the values we have: \[ |A \cap H \text{ only}| = 6 - 3 = 3 \] So, **3 students are taking History and Anthropology but not Psychology.** ### Summary of Answers: (a) 31 students are taking only Anthropology. (b) 45 students are taking Anthropology or History. (c) 3 students are taking History and Anthropology but not Psychology.