To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?
Options:
(A) 30,000
(B) 2,70,000
(C) 3,90,000
(D) 4,20,000
Official Answer Key:
D
Step-by-Step Solution & Explanation
Work in thousands. Let x = |A∩B∩C| = number answering none.
Inclusion–exclusion: |A∪B∪C| = 330 + 250 + 260 − 100 − 90 − 80 + x = 570 + x.
Total = |A∪B∪C| + none ⇒ 630 = (570 + x) + x ⇒ x = 30 (i.e. 30,000).
Exactly two correct = (100−30) + (90−30) + (80−30) = 70 + 60 + 50 = 180.
Passed = exactly two + all three = 180 + 30 = 210 ⇒ 2,10,000.
Failed = 630 − 210 = 420 ⇒ 4,20,000.
Distractor (C) 3,90,000 arises from forgetting to subtract the all-three group from each pairwise count.
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