Question Statement:
How many 3-digit numbers can be formed using three distinct single digit prime
numbers?
Options:
-
(A) 64
-
(B) 24
-
(C) 12
-
(D) 4
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.4)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Permutations of Prime Digits
- The single-digit prime numbers are: $\{2, 3, 5, 7\}$ (exactly 4 distinct primes).
- To form a 3-digit number using three distinct single-digit primes, we choose 3 primes from 4 and arrange them:
$N = P(4, 3) = \frac{4!}{(4-3)!} = 4 \times 3 \times 2 = 24$
**2. Option Analysis**:
- **Option (A) [12] is INCORRECT**: Undercounts possible arrangements.
- **Option (B) [24] is CORRECT**: Exactly $4 \times 3 \times 2 = 24$ distinct 3-digit numbers can be constructed.
- **Option (C) [36] is INCORRECT**: Exceeds available permutations.
- **Option (D) [48] is INCORRECT**: Assumes replacement or invalid digits.
**3. Final Answer**:
- **Correct Choice**: **B**
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