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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?

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? Correct Answer 4,20,000

To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly.

So, a candidate who answer only one or didn’t answer any question will fail the test.

Number of candidates who answered question A correctly, n(A)= 3,30,000

Number of candidates who answered question B correctly, n(B) = 2,50,000

Number of candidates who answered question C correctly, n(C) = 2,60,000

Number of candidates who answered both question A and B correctly, n(A ⋂ B) = 1,00,000

Number of candidates who answered both question B and C correctly, n(B ⋂ C) = 90,000

Number of candidates who answered both question A and C correctly, n(A ⋂ C) = 80,000

Let the number of candidates who answered all the three questions n(A ⋂ B ⋂ C) = x

Given that, the number of students answering all questions correctly is the same as the number answering none.

So, the number of candidates answered zero questions = x

The number of candidates who answered only question A and B correctly = n(A ⋂ B) - n(A ⋂ B ⋂ C) = 1,00,000 – x

The number of candidates who answered only question B and C correctly = n(B ⋂ C) - n(A ⋂ B ⋂ C) = 90,000 – x

The number of candidates who answered only question A and C correctly = n(A ⋂ C) - n(A ⋂ B ⋂ C) = 80,000 – x

The number of candidates who answered only question A correctly = n(A) - n(A ⋂ B) - n(A ⋂ C) + n(A ⋂ B ⋂ C) = 3,30,000 - 1,00,000 - 80,000 + x = 1,50,000 + x

The number of candidates who answered only question A correctly = n(B) - n(A ⋂ B) - n(B ⋂ C) + n(A ⋂ B ⋂ C) = 2,50,000 - 1,00,000 - 90,000 + x = 60,000 + x

The number of candidates who answered only question A correctly = n(C) - n(B ⋂ C) - n(A ⋂ C) + n(A ⋂ B ⋂ C) = 2,60,000 - 90,000 - 80,000 + x = 90,000 + x

Given that, total number of candidates appeared for the test = 6,30,000

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From the Venn diagram,

Total number of candidates appeared for the test = (1,50,000 + x) + (60,000 + x) + (90,000 + x) + (1,00,000 – x) + (90,000 – x) + (80,000 – x) + x + x

= 5,70,000 + 2x

⇒ 5,70,000 + 2x = 6,30,000

⇒ x = 30,000

The number candidates who answered all the three questions correctly = The number of questions who answered zero questions correctly = x = 30,000

The number of candidates failed to clear the test = Number of students who answered only one question correctly + number of students who answered zero questions correctly

= 1,50,000 + x + 60,000 + x + 90,000 + x + x

= 3,00,000 + 4x

= 3,00,000 + 4(30,000)

= 4,20,000

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