A class has 175 learners. The given table describes the number of learners studying one or more of the subsequent subjects in this case

                Subjects                                   No. of students

                Mathematics                                  100

                Physics                                           70

                Chemistry                                       46

                Mathematics and Physics               30

                Mathematics and Chemistry          28

                Physics and Chemistry                 23

                Mathematics, Physics and Chemistry      18

 How many learners are study Physics alone, Mathematics alone and Chemistry alone? Are there learners who have not taken any one of these subjects?

Ans)  Let P, C, M shows the sets of learners taking Physics, Chemistry and Mathematics correspondingly.

Let a, b, c, d, e, f, g represents the number of students defined in the bounded region as presented in the diagram then

                a + d + e + g  = 70

                c + d + f + g = 100

                b + e + f + g = 46

                d + g = 30

                e + g = 23

                f + g = 28

                g = 18   

After measuring we get g = 18, f = 10, e = 5, d = 12, a = 35, b = 13 and c = 60

a + b + c + d + e + f + g = 153

So, the number of learners who have not studied any of these 3 subjects

                = 175 -153 = 22

                Total number of learners studying Mathematics only, c = 60

                Total number of learners studying Physics only, a = 35

                Total number of learners studying Chemistry only, b = 13.