Question 9 A group of 85 students were surveyed about the courses they were taking at their college with the following results: 52 students said they were taking Math. 56 students said they were taking English. 49 students said they were taking History. 32 students said they were taking Math and English. 30 students said they were taking Math and History. 33 students said they were taking English and History. 19 students said they were taking all three courses. How many students took none of the courses? Question Help: Vint Submit Question

To find out how many students took none of the courses, we can use the principle of inclusion-exclusion. Let: - \( M \) = number of students taking Math = 52 - \( E \) = number of students taking English = 56 - \( H \) = number of students taking History = 49 - \( ME \) = number of students taking both Math and English = 32 - \( MH \) = number of students taking both Math and History = 30 - \( EH \) = number of students taking both English and History = 33 - \( MEH \) = number of students taking all three courses = 19 Using the principle of inclusion-exclusion, the total number of students taking at least one course is given by: \[ |M \cup E \cup H| = |M| + |E| + |H| - |ME| - |MH| - |EH| + |MEH| \] Substituting the values: \[ |M \cup E \cup H| = 52 + 56 + 49 - 32 - 30 - 33 + 19 \] Calculating step-by-step: 1. Sum of students taking each course: \[ 52 + 56 + 49 = 157 \] 2. Sum of students taking two courses: \[ 32 + 30 + 33 = 95 \] 3. Now, substitute these sums into the inclusion-exclusion formula: \[ |M \cup E \cup H| = 157 - 95 + 19 \] 4. Calculate: \[ 157 - 95 = 62 \] \[ 62 + 19 = 81 \] So, \( |M \cup E \cup H| = 81 \). Now, to find the number of students who took none of the courses, we subtract the number of students taking at least one course from the total number of students surveyed: \[ \text{Number of students taking none} = \text{Total students} - |M \cup E \cup H| \] \[ \text{Number of students taking none} = 85 - 81 = 4 \] Thus, the number of students who took none of the courses is **4**.