in a school of 100 students, 59 are taking math, 51 are in english, 35 are in history, 24 are in both math and english, 19 are in both english and history, 13 in both math and history, and 11 in all three courses. how many students are taking:

a) none of the classes?
b) only math?
c) history and english?
d) only english?
e) exactly two of the three classes?

I drew three overlapping circles.

all three =11

math and history only 13-11 = 2
english and history only 19-11 = 8
math and english only 24-11 = 13

so math only = 59-13-11-2 = 33
english only = 51-13-11-8 = 19
history only = 35-2-11-8 = 14

add them all up and you get 100, so no students are taking none.

To solve this problem, we can use the principle of inclusion-exclusion. Let's break down the problem step by step:

a) To find the number of students taking none of the classes, we need to subtract the total number of students taking at least one class from the total number of students in the school.

To calculate the total number of students taking at least one class, we add the number of students taking each individual course (59 + 51 + 35). But since we have counted some students multiple times, we need to subtract the students who are taking two courses (24 + 19 + 13). Finally, we need to add back the students who are taking all three courses (11).

Total students taking at least one class = 59 + 51 + 35 - 24 - 19 - 13 + 11 = 100.

Therefore, the number of students taking none of the classes is 100 - 100 = 0.

b) To find the number of students taking only math, we need to subtract the students taking both math and another course (math and English, math and history) from the total number of students taking math.

Number of students taking only math = Students taking math - (Students taking both math and English + Students taking both math and history - Students taking all three courses).

Number of students taking only math = 59 - (24 + 13 - 11) = 59 - 26 = 33.

Therefore, the number of students taking only math is 33.

c) To find the number of students taking history and English, we need to subtract the students taking only one of the courses (history or English) from the total number of students taking both history and English.

Number of students taking history and English = Students taking both history and English - (Students taking only history + Students taking only English - Students taking all three courses).

Number of students taking history and English = 11 - (13 - 11 + 19 - 11) = 11 - 10 = 1.

Therefore, the number of students taking history and English is 1.

d) To find the number of students taking only English, we can follow a similar approach as in part b.

Number of students taking only English = Students taking English - (Students taking both math and English + Students taking both English and history - Students taking all three courses).

Number of students taking only English = 51 - (24 + 19 - 11) = 51 - 32 = 19.

Therefore, the number of students taking only English is 19.

e) To find the number of students taking exactly two of the three classes, we need to subtract the students taking all three courses from the total number of students taking at least two courses.

Number of students taking exactly two of the three classes = Students taking at least two courses - Students taking all three courses.

Number of students taking at least two courses = Students taking both math and English + Students taking both English and history + Students taking both math and history - Students taking all three courses.

Number of students taking at least two courses = 24 + 19 + 13 - 11 = 45.

Number of students taking exactly two of the three classes = 45 - 11 = 34.

Therefore, the number of students taking exactly two of the three classes is 34.

In summary:

a) None of the classes: 0.
b) Only math: 33.
c) History and English: 1.
d) Only English: 19.
e) Exactly two of the three classes: 34.