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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?
  • a)
    30,000
  • b)
    2,70,000
  • c)
    3,90,000
  • d)
    4,20,000
Correct answer is option 'D'. Can you explain this answer?
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Solution:

Given data:
- Total candidates appeared for the test = 6,30,000
- Question A answered correctly = 3,30,000
- Question B answered correctly = 2,50,000
- Question C answered correctly = 2,60,000
- Questions A and B both answered correctly = 1,00,000
- Questions B and C both answered correctly = 90,000
- Questions A and C both answered correctly = 80,000
- Number of students answering all questions correctly = Number of students answering none.

We can use the principle of inclusion-exclusion to find the number of candidates who failed to clear the test.

Let's find the number of candidates who answered at least one question correctly:
- Number of candidates who answered question A correctly = 3,30,000
- Number of candidates who answered question B correctly = 2,50,000
- Number of candidates who answered question C correctly = 2,60,000
- Number of candidates who answered both questions A and B correctly = 1,00,000
- Number of candidates who answered both questions B and C correctly = 90,000
- Number of candidates who answered both questions A and C correctly = 80,000

Number of candidates who answered at least one question correctly = (330000 + 250000 + 260000) - (100000 + 90000 + 80000) + number of candidates who answered all questions correctly

Simplifying the above equation, we get:
Number of candidates who answered at least one question correctly = 630000 - 110000 + number of candidates who answered all questions correctly

We are given that the number of candidates who answered all questions correctly is the same as the number of candidates who answered none. Therefore,
Number of candidates who answered at least one question correctly = 630000 - 110000 + number of candidates who answered none

To find the number of candidates who answered none, we subtract the number of candidates who answered at least one question correctly from the total number of candidates:
Number of candidates who answered none = 630000 - (630000 - 110000 + number of candidates who answered none)
Number of candidates who answered none = 110000 - number of candidates who answered none

Solving the above equation, we get:
2 * number of candidates who answered none = 110000
Number of candidates who answered none = 55,000

Now, we can find the number of candidates who failed to clear the test:
Number of candidates who failed to clear the test = Total candidates - Number of candidates who answered at least one question correctly
Number of candidates who failed to clear the test = 630000 - (630000 - 110000 + 55000)
Number of candidates who failed to clear the test = 420,000

Therefore, the correct option is (D) 4,20,000.
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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?a)30,000b)2,70,000c)3,90,000d)4,20,000Correct answer is option 'D'. Can you explain this answer?
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