Question

among 100 students 32 studey maths 20 study physics, 45 study biology, 15 study maths and biology , 7 study maths and physics,10 study physics and biology and 30 do not study any of the three subjects.

Answer 1

so what's the question

Answer 2

i) Number of students studying all three subjects = 70

ii) Number of students studying exactly one of the three subjects = 48

Given,

• Total students = n(U) = 100

• Students who study Mathematics = n(M) = 32

• Students who study Physics = n(P) = 20

• Students who study Biology = n(B) = 45

• Students who study both Mathematics and Biology = n(M∩B) = 15

• Students who study both Mathematics and Physics = n(M∩P) = 7

• Students who study both Physics and Biology = n(P∩B) = 10

• Students who do not study any of the three subjects = n(M∪P∪B)' = 30
• We know that,

           n(U) = n(M∪P∪B) + n(M∪P∪B)'

            100 = n(M∪P∪B) + 30

           n(M∪P∪B) = 100 - 30 = 70

• Number of students studying all the three subjects = n(M∩P∩B)

              = n(M∪P∪B) - n(M) - n(P) -n(B) + n(M∩B) + n(M∩P) + n(P∩B)

              = 70 - 32 - 20 - 45 + 15 + 7 + 10

              = 5

•  Number of students studying only Mathematics

               = n(M) - n(M∩B) - n(M∩P) + n(M∩P∩B)

               = 32 - 15 - 7 + 5 = 15

• Number of students studying only Physics

               = n(P) - n(P∩B) - n(M∩P) + n(M∩P∩B)

               = 20 - 7 - 10 + 5 = 8

• Number of students studying only Biology

               = n(B) - n(P∩B) - n(M∩B) + n(M∩P∩B)

               = 45 - 10 - 15 + 5 = 25

• Therefore, number of studying exactly one of the three subjects

               = 15 + 8 + 25 = 48