Problem Setup: Candidates and Passing Criteria

We are given the total number of candidates and the number of candidates who answered specific questions correctly. The passing threshold is answering at least 2 out of 3 questions correctly.

• Total Candidates ($N$): $6,30,000$
• Candidates answering A correctly ($|A|$): $3,30,000$
• Candidates answering B correctly ($|B|$): $2,50,000$
• Candidates answering C correctly ($|C|$): $2,60,000$
• Candidates answering A and B correctly ($|A \cap B|$): $1,00,000$
• Candidates answering B and C correctly ($|B \cap C|$): $90,000$
• Candidates answering A and C correctly ($|A \cap C|$): $80,000$
• Let $x$ be the number of candidates answering all three questions correctly ($|A \cap B \cap C|$).
• We are given that the number answering all correctly is equal to the number answering none correctly. So, number answering none ($|(A \cup B \cup C)'|$) is also $x$.

Calculate Number Answering All/None

Using the Principle of Inclusion-Exclusion for three sets:

$|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |A \cap C|) + |A \cap B \cap C|$

Substitute the given values:

$|A \cup B \cup C| = 3,30,000 + 2,50,000 + 2,60,000 - (1,00,000 + 90,000 + 80,000) + x$

$|A \cup B \cup C| = 8,40,000 - 2,70,000 + x$

$|A \cup B \cup C| = 5,70,000 + x$

The total number of candidates is the sum of those in the union ($A \cup B \cup C$) and those outside the union (none):

$N = |A \cup B \cup C| + |(A \cup B \cup C)'|$

$6,30,000 = (5,70,000 + x) + x$

$6,30,000 = 5,70,000 + 2x$

$2x = 6,30,000 - 5,70,000$

$2x = 60,000$

$x = 30,000$

So, $30,000$ candidates answered all questions correctly, and $30,000$ answered none correctly.

Calculate Number Passed

Candidates pass if they answer at least 2 questions correctly. This includes those who answered exactly 2 correctly and those who answered exactly 3 correctly.

• Exactly 3 correct: This is $|A \cap B \cap C| = x = 30,000$.
• Exactly 2 correct: This is calculated by taking the intersections of two sets and subtracting those who got all three correct.
  • A and B only: $|A \cap B| - |A \cap B \cap C| = 1,00,000 - 30,000 = 70,000$
  • B and C only: $|B \cap C| - |A \cap B \cap C| = 90,000 - 30,000 = 60,000$
  • A and C only: $|A \cap C| - |A \cap B \cap C| = 80,000 - 30,000 = 50,000$
  Total with exactly 2 correct = $70,000 + 60,000 + 50,000 = 1,80,000$.

Total candidates who passed = (Exactly 2 correct) + (Exactly 3 correct)

Total Passed = $1,80,000 + 30,000 = 2,10,000$.

Calculate Number Failed

The number of candidates who failed is the total number of candidates minus the number who passed.

Failed Candidates = Total Candidates - Total Passed

Failed Candidates = $6,30,000 - 2,10,000$

Failed Candidates = $4,20,000$