Question 8 A group of 461 students were surveyed about the courses they were taking at their college with the following results: 220 students said they were taking Math. 193 students said they were taking English. 236 students said they were taking History. 60 students said they were taking Math and English. 127 students said they were taking Math and History. 91 students said they were taking English and History. 37 students said they were taking all three courses. How many students took English and History, but not Math? Question Help: Vint Submit Question

To find the number of students who took English and History but not Math, we can use the principle of inclusion-exclusion and the information provided. Let's define the following: - Let \( M \) be the set of students taking Math. - Let \( E \) be the set of students taking English. - Let \( H \) be the set of students taking History. From the information given: - \( |M| = 220 \) - \( |E| = 193 \) - \( |H| = 236 \) - \( |M \cap E| = 60 \) - \( |M \cap H| = 127 \) - \( |E \cap H| = 91 \) - \( |M \cap E \cap H| = 37 \) We want to find the number of students who took English and History but not Math, which can be represented as \( |E \cap H \cap M'| \), where \( M' \) is the complement of set \( M \). Using the formula for the intersection of two sets: \[ |E \cap H| = |E \cap H \cap M| + |E \cap H \cap M'| \] We can rearrange this to find \( |E \cap H \cap M'| \): \[ |E \cap H \cap M'| = |E \cap H| - |E \cap H \cap M| \] We know: - \( |E \cap H| = 91 \) - \( |E \cap H \cap M| = |M \cap E \cap H| = 37 \) Now we can substitute these values into the equation: \[ |E \cap H \cap M'| = 91 - 37 = 54 \] Thus, the number of students who took English and History but not Math is **54**.