Question

In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is

• A. 6
• B. 9
• C. 7
• D. All of these

Answer 1

Answer: (C)

7

Answer 2

n(M) = 23, n(P) = 24, n(3)= 19

n(M ∩ P) = 12, n(M ∩ C)= 9, n(P ∩ C)= 7

n(M ∩ P ∩ C) = 4

We have to find n(M ∩ P′ ∩ C′), n(P ∩ M ′ ∩ C′ ), n ( C ∩ M ′ ∩ P ′)

Now n (M ∩ P′ ∩ C′) = n[M ∩ (P ∪ C)′]

= n(M)– n(M ∩ (P ∪ C)) $= n ( M ) - n [ ( M \cap P ) \cup ( M \cap C ) ]$

= n(M) – n(M ∩ P)– n(M ∩ C) + n(M ∩ P ∩ C)

= 23 –12 – 9 + 4 = 27 –21 = 6

n(P ∩ M′ ∩ C′) = n[P ∩ (M ∪ C)′]

= n(P)– n[P ∩ (M ∪ C)] = $n ( P ) - n [ ( P \cap M ) \cup ( P \cap C ) ]$ = n(P) – n(P ∩ M) – n(P ∩ C) + n(P ∩ M ∩ C)

= 24 – 12 – 7 + 4 = 9

n(C ∩ M′ ∩ P′) = n(3) – n(C ∩ P) – n(C ∩ M)+ n(C ∩ P ∩ M) = 19 – 7 – 9 + 4 = 23 – 16 = 7

Hence (3) is the correct answer.