Explanation:
150 students overall.
an immediate assumption would be that these 150 students are the universal set of students taking these classes.
but we have to be careful and check. could it be that there are more than these 3 classes for the 150 students ? or that there is an option to be a student and not take any of the classes ?
the last question (c) suggests something like that.
so, these 150 students are the universal set for all these cases.
the sum of all students that took 2 classes contains also the ones that took all 3 classes. each of these students (that took all 3 classes) are then counted three times in these groups - one time for each pair of classes.
so, 26 + 28 + 32 - 22 - 22 = 42 is the number of students that took at least 2 classes (the subtraction of 2×22 eliminates two of the triple counting of the students that took all 3 classes).
since these 42 students contain also the ones with 3 courses, we need to subtract the 22 with 3 classes a third time again to get the number of students that had exactly 2 classes :
42 - 22 = 20.
so, we have 22 students attending all 3 class.
then 20 more attending exactly 2 classes.
in detail it means for these 20
26 - 22 = 4 had taken exactly math and physics.
28 - 22 = 6 had taken exactly math and chemistry.
32 - 22 = 10 had taken exactly chemistry and physics.
and that means for the individual classes
84 - 22 - 4 - 6 = 52 had taken math only.
64 - 22 - 6 - 10 = 26 had taken chemistry only.
48 - 22 - 4 - 10 = 12 had taken physics only.
in total
150 - 52 - 26 - 12 - 20 - 22 = 18
18 students of these 150 have not attended any of these 3 classes.
(a)
so,
52 + 26 + 12 = 90
90 students took one course only.
(b)
so, the number of students that had at most 2 classes is the number of students with exactly one class (90) and exactly 2 classes (20) AND the ones without any of these 3 classes : 90 + 20 + 18 = 128
128 students took at most 2 courses.
we could have gotten this also by saying all the students not taking all 3 classes : 150 - 22 = 128
(c)
the original assumption was that 150 students did at least one of the classes. but that was wrong.
we found that 18 students did not attend any of the classes.
that means that
150 - 18 = 132
132 students took at least one of the courses.