In a group of students, 10 students like Mathematics, 12 students like English,
4 students like both Mathematics and English, and 6 students like neither
Mathematics nor English. The number of students in the group is ____
Options:
(A) 18
(B) 20
(C) 24
(D) 32
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.5)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given data about a group of students and their subject preferences:
- Total students who like Mathematics, $n(M) = 10$
- Total students who like English, $n(E) = 12$
- Students who like both Mathematics and English, $n(M \cap E) = 4$
- Students who like neither Mathematics nor English, $n((M \cup E)^c) = 6$
We need to find the total number of students in the group, denoted by universal set size $|U|$.
**2. Step-by-Step Mathematical Derivation**:
1. **Principle of Inclusion-Exclusion**:
The number of students who like at least one of the subjects is:
$n(M \cup E) = n(M) + n(E) - n(M \cap E)$
Substitute the given values:
$n(M \cup E) = 10 + 12 - 4 = 18$
2. **Total Population Calculation**:
The total number of students consists of those who like at least one subject plus those who like neither:
$|U| = n(M \cup E) + n((M \cup E)^c)$
$|U| = 18 + 6 = 24$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 18**: ❌ **INCORRECT**. This is $n(M \cup E)$, forgetting the 6 students who like neither.
- **(B) 20**: ❌ **INCORRECT**. Result of incorrect subtraction or missing terms.
- **(C) 24**: ✅ **CORRECT**. Accurately computed as $(10 + 12 - 4) + 6 = 24$.
- **(D) 32**: ❌ **INCORRECT**. Result of simply summing all numbers without inclusion-exclusion: $10 + 12 + 4 + 6 = 32$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Always draw or visualize a 2-set Venn diagram: Only Math $= 10 - 4 = 6$, Only English $= 12 - 4 = 8$, Both $= 4$, Neither $= 6$. Total $= 6 + 8 + 4 + 6 = 24$.
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