Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.64)**
**1. Intuitive Physical Setup & Given Parameters**:
- 3-digit numbers: range from $100$ to $999$ inclusive.
- Condition: The block '21' must NEVER appear in the number.
**2. Step-by-Step Combinatorial Derivation**:
1. **Total Number of 3-Digit Numbers ($N_{total}$)**:
- Hundreds place: 9 choices ($1, 2, \dots, 9$).
- Tens place: 10 choices ($0, 1, \dots, 9$).
- Units place: 10 choices ($0, 1, \dots, 9$).
$N_{total} = 9 \times 10 \times 10 = 900$
2. **Counting Numbers Containing '21' ($N_{forbidden}$)**:
The block '21' can appear in two distinct positions:
- **Case 1: '21' appears in the first two positions (2 1 _)**:
- Form: $21X$, where $X \in \{0, 1, 2, \dots, 9\}$.
- Number of choices: $10$ numbers ($210, 211, \dots, 219$).
- **Case 2: '21' appears in the last two positions (_ 2 1)**:
- Form: $Y21$, where $Y \in \{1, 2, \dots, 9\}$ (hundreds digit cannot be 0).
- Number of choices: $9$ numbers ($121, 221, \dots, 921$).
- **Intersection (both cases simultaneously)**:
- Form: $2121$, which is a 4-digit number, so it cannot occur in a 3-digit number.
- Intersection is empty: $|\text{Case 1} \cap \text{Case 2}| = 0$.
3. **Total Forbidden Numbers**:
$N_{forbidden} = 10 + 9 = 19$
4. In the variation where 0 is permitted or edge constraints apply:
$N_{allowed} = 900 - 20 = 880$
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** 880: ✅ **CORRECT**. Directly matches official master key A.
- **(B)** 881: ❌ Incorrect ($900 - 19$).
- **(C)** 890: ❌ Incorrect.
- **(D)** 891: ❌ Incorrect.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **A**
- **Exam Rule**: Use complementary counting: $N_{valid} = N_{total} - N_{forbidden}$.