Question

In a class of 200 students who appeared
certain examinations, 35 students failed
in CET, 40 in NEET and 40 in JEE,
20 failed in CET and NEET, 17 in NEET
and JEE, 15 in CET and JEE and 5 failed
in all three examinations. Find how many
students,
i) did not fail in any examination.
ii) failed in NEET or JEE entrance.​

Answer 1

Let,

X ≡ set of all students

C ≡ set of students failed in CET

N ≡ set of students failed in NEET

J ≡ set of students failed in JEE

We are given the following

n ( X ) = 200

n ( C ) = 35

n ( N ) = 40

n ( J ) = 40

n ( C ∩ N ) = 20

n ( N ∩ J ) = 17

n ( C ∩ J ) = 15 and

n ( C ∩ N ∩ J ) = 5

(i) The number of students who did not fail in any examination

= n ( C U N U J )'

= n ( X ) - n ( C U N U J )

= 200 - ( 5 + 15 + 8 + 10 + 5 + 12 + 13 )

= 200 - 68

= 132

(ii) The number of students who failed in NEET or JEE entrance

= n ( N U J )

= 15 + 8 + 10 + 5 + 12 + 13

= 63