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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?

  1. $30,000$
  2. $2,70,000$
  3. $3,90,000$
  4. $4,20,000$
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1 Answer

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12 votes
Best answer
Answer D)

Say there are x students who answered all correctly

So, according to condition there are x student who answered none

Equation using inclusion exclusion principle

$630000-x=330000+250000+260000-100000-90000-80000+x$

$x=30000$

Now, it is given that to pass u need to answer at least 2 out of 3 questions

So, those who answered one answer correctly also considered to be fail

Only A answered correctly$330000-100000-80000+30000=180000$

Only B answered correctly $250000-100000-90000+30000=90000$

Only C answered correctly $260000-80000-90000+30000=120000$

So, total failed student $180000+90000+120000+30000=420000$
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4 Comments

@Chidesh1397 @mohan123   This is because LHS indicates the number of students who have answered at least one question correctly, which is equal to total number of students  $-$  number of students who have answered all questions correctly , which is equal to  total number of students  $-$  number of students who have not answered not a single question correctly(using the condition mentioned in the question).
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For ease of calculation, we can remove four zeroes from each number.
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Yes, that’s for the exam but here she has to write it, otherwise would look incomplete.
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Answer:

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