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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. $4,20,000$

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 $180000+90000+120000+30000=420000$
by

@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).
For ease of calculation, we can remove four zeroes from each number.
Yes, that’s for the exam but here she has to write it, otherwise would look incomplete.