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In a class of 200 students who appeared in certain examinations, 35 students faded in CET, 40 in NEET and 40 in JEE, 20 faded in CET and NEET, 17 in NEET and JEE, 15 in CET and JEE and 5 faded in ad three examinations. Find how many students

(i) did not fail in any examination.

(ii) faded in NEET or JEE entrance.

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Let A = set of students who failed in CET

B = set of students who failed in NEET

C = set of students who failed in JEE

X = set of all students

∴ n(X) = 200, n(A) = 35, n(B) = 40, n(C) = 40,

n(A ∩ B) = 20, n(B ∩ C) = 17, n(A ∩ C) = 15,

n(A ∩ B ∩ C) = 5

(i) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)

= 35 + 40 + 40 – 20 – 17 – 15 + 5

= 68

∴ No. of students who did not fail in any exam

= n(X) – n(A ∪ B ∪ C)

= 200 – 68

= 132

(ii) No. of students who failed in NEET or JEE entrance = n(B ∪ C)

= n(B) + n(C) – n(B ∩ C)

= 40 + 40 – 17

= 63

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