Question 16 A group of 574 students were surveyed about the courses they were taking at their college with the following results: 284 students said they were taking Math. 244 students said they were taking English. 314 students said they were taking History. 110 students said they were taking Math and English. 154 students said they were taking Math and History. 155 students said they were taking English and History. 71 students said they were taking all three courses. Draw a Venn diagram and use it to answer the question. How many students took English and History, but not Math? Question Help: Video Submit Question

To solve the problem, we can use a Venn diagram to represent the relationships between the three courses: Math (M), English (E), and History (H). Let's denote: - \( n(M) = 284 \) (students taking Math) - \( n(E) = 244 \) (students taking English) - \( n(H) = 314 \) (students taking History) - \( n(M \cap E) = 110 \) (students taking both Math and English) - \( n(M \cap H) = 154 \) (students taking both Math and History) - \( n(E \cap H) = 155 \) (students taking both English and History) - \( n(M \cap E \cap H) = 71 \) (students taking all three courses) ### Step 1: Calculate the number of students in each section of the Venn diagram. 1. **Students taking all three courses (M, E, H)**: - \( n(M \cap E \cap H) = 71 \) 2. **Students taking Math and English, but not History**: - \( n(M \cap E) - n(M \cap E \cap H) = 110 - 71 = 39 \) 3. **Students taking Math and History, but not English**: - \( n(M \cap H) - n(M \cap E \cap H) = 154 - 71 = 83 \) 4. **Students taking English and History, but not Math**: - \( n(E \cap H) - n(M \cap E \cap H) = 155 - 71 = 84 \) 5. **Students taking only Math**: - \( n(M) - (n(M \cap E) + n(M \cap H) - n(M \cap E \cap H)) \) - \( = 284 - (110 + 154 - 71) = 284 - 193 = 91 \) 6. **Students taking only English**: - \( n(E) - (n(M \cap E) + n(E \cap H) - n(M \cap E \cap H)) \) - \( = 244 - (110 + 155 - 71) = 244 - 194 = 50 \) 7. **Students taking only History**: - \( n(H) - (n(M \cap H) + n(E \cap H) - n(M \cap E \cap H)) \) - \( = 314 - (154 + 155 - 71) = 314 - 238 = 76 \) ### Step 2: Summarize the counts in the Venn diagram. - Only Math: 91 - Only English: 50 - Only History: 76 - Math and English, but not History: 39 - Math and History, but not English: 83 - English and History, but not Math: 84 - All three courses: 71 ### Step 3: Answer the question. The number of students who took English and History, but not Math is **84**.