Discussion
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?
Answer: D
The number who failed is 4,20,000.
Work in thousands. Let \(x\) be the number answering all three correctly, which equals the number answering none. By inclusion-exclusion, the number answering at least one is
\[ \begin{aligned} &330 + 250 + 260 - 100 - 90 - 80 + x \\ &= 570 + x \end{aligned} \]
so the number answering none is \(630 - 570 - x = 60 - x\). Setting this equal to \(x\) gives \(x = 30\). Exactly two correct: \((100 + 90 + 80) - 3x = 180\). Those who pass have two or three correct: \(180 + 30 = 210\). So \(630 - 210 = 420\) thousand fail.
30,000 is the number answering all (or none), and 2,70,000 is the sum of the pairwise counts, which double-counts those answering all three.
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