Question Statement:
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
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?
Options:
-
(A) 30,000
-
(B) 2,70,000
-
(C) 3,90,000
-
(D) 4,20,000
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.08)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Set Theory & Venn Diagrams (Inclusion-Exclusion Principle)
**2. Detailed Step-by-Step Derivation**:
- Total candidates $U = 6,30,000$.
- Let $A, B, C$ denote the sets of candidates correctly answering questions A, B, and C respectively:
- $n(A) = 3,30,000$
- $n(B) = 2,50,000$
- $n(C) = 2,60,000$
- $n(A \cap B) = 90,000$
- $n(A \cap C) = 80,000$
- Let $n(B \cap C) = y$
- Let the number of candidates answering all 3 questions correctly be $n(A \cap B \cap C) = x$.
- Given that the number answering none is also equal to $x$:
$n( ext{None}) = U - n(A \cup B \cup C) = x \implies n(A \cup B \cup C) = U - x = 6,30,000 - x$
- By the Principle of Inclusion-Exclusion:
$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$
$6,30,000 - x = 3,30,000 + 2,50,000 + 2,60,000 - 90,000 - 80,000 - y + x$
$6,30,000 - x = 6,70,000 - y + x$
$y - 2x = 6,70,000 - 6,30,000 = 40,000 \implies y - x = x + 40,000$
- A candidate passes if they answer **at least 2 out of 3** questions correctly.
- Number of candidates passing = candidates answering exactly 2 questions + candidates answering all 3 questions:
$N_{ ext{pass}} = [n(A \cap B) - x] + [n(A \cap C) - x] + [n(B \cap C) - x] + x$
$N_{ ext{pass}} = (90,000 - x) + (80,000 - x) + (y - x) + x = 1,70,000 + y - 2x$
Substitute $(y - 2x) = 40,000$:
$N_{ ext{pass}} = 1,70,000 + 40,000 = 2,10,000$
- Therefore, the number of candidates who **failed** to clear the test:
$N_{ ext{fail}} = U - N_{ ext{pass}} = 6,30,000 - 2,10,000 = 4,20,000$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: 30,000 is an unrelated sub-count.
- **Option (B) [INCORRECT]**: 2,70,000 does not correctly account for failing candidates.
- **Option (C) [INCORRECT]**: 3,90,000 represents arithmetic without properly applying the inclusion-exclusion constraint.
- **Option (D) [CORRECT]**: 4,20,000 accurately represents the total failed candidates ($6,30,000 - 2,10,000$).
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).