Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.4)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to find the number of 3-digit numbers that can be formed using **three distinct single-digit prime numbers**.
**2. Step-by-Step Mathematical Derivation**:
1. **Identify the Single-Digit Prime Numbers**:
The single-digit primes (base 10) are:
\mathcal{P} = \{2, 3, 5, 7\}
There are exactly $n = 4$ such primes: 2, 3, 5, and 7. Note that none of them is 0, so any combination of three distinct digits from this set can form a valid 3-digit number in any position (hundreds, tens, units).
2. **Count the Permutations**:
We need to select 3 distinct digits out of the 4 available primes and arrange them in order to form a 3-digit number.
The number of such arrangements is given by the permutation formula $P(n, r)$:
P(4, 3) = \frac{4!}{(4-3)!} = \frac{4 \times 3 \times 2 \times 1}{1} = 24
Alternatively:
- Ways to pick the hundreds digit: 4 choices
- Ways to pick the tens digit: 3 choices
- Ways to pick the units digit: 2 choices
\text{Total 3-digit numbers} = 4 \times 3 \times 2 = 24
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 64**: ❌ **INCORRECT**. Corresponds to allowing repetition ($4^3 = 64$), but the problem explicitly states "distinct".
- **(B) 24**: ✅ **CORRECT**. Exactly $P(4, 3) = 4 \times 3 \times 2 = 24$ distinct numbers.
- **(C) 12**: ❌ **INCORRECT**. Half the correct value, often from mistakenly dividing by 2.
- **(D) 4**: ❌ **INCORRECT**. Represents combinations $\binom{4}{3} = 4$ without permuting the digit positions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Distinguish between combinations (subsets) and permutations (numbers where digit order matters). Since order signifies place value, multiply combinations by $3! = 6$, yielding $4 \times 6 = 24$.