Question 7 A group of 498 students were surveyed about the courses they were taking at their college with the following results: 207 students said they were taking Math. 266 students said they were taking English. 280 students said they were taking History. 118 students said they were taking Math and English. 121 students said they were taking Math and History. 162 students said they were taking English and History. 100 students said they were taking all three courses. How many students took Math, English, or History? Question Help: Hint Submit Question

To find the number of students who took at least one of the courses (Math, English, or History), we can use the principle of inclusion-exclusion. Let: - \( M \) = number of students taking Math = 207 - \( E \) = number of students taking English = 266 - \( H \) = number of students taking History = 280 - \( ME \) = number of students taking both Math and English = 118 - \( MH \) = number of students taking both Math and History = 121 - \( EH \) = number of students taking both English and History = 162 - \( MEH \) = number of students taking all three courses = 100 Using the inclusion-exclusion principle, the formula for the number of students taking at least one of the courses is: \[ |M \cup E \cup H| = |M| + |E| + |H| - |ME| - |MH| - |EH| + |MEH| \] Substituting the values: \[ |M \cup E \cup H| = 207 + 266 + 280 - 118 - 121 - 162 + 100 \] Calculating step-by-step: 1. Sum of students taking each course: \[ 207 + 266 + 280 = 753 \] 2. Sum of students taking two courses: \[ 118 + 121 + 162 = 401 \] 3. Now substitute these sums into the formula: \[ |M \cup E \cup H| = 753 - 401 + 100 \] 4. Calculate: \[ 753 - 401 = 352 \] \[ 352 + 100 = 452 \] Thus, the number of students who took Math, English, or History is **452**.