Number of 3 digit numbers that can be formed having unit digit as zero and repetition of digit is allowed is

Options

A. 72
B. 81
C. 100
D. 90

Answer

To solve the problem of finding the number of 3-digit numbers that can be formed with the unit digit as zero and where repetition of digits is allowed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of the 3-Digit Number**:
A 3-digit number can be represented as \( ABC \), where \( A \) is the hundreds place, \( B \) is the tens place, and \( C \) is the units place. According to the problem, the unit digit \( C \) must always be 0.

2. **Filling the Unit Place**:
Since the unit digit \( C \) is fixed as 0, there is only 1 way to fill this place:
\[ C = 0 \quad \text{(1 way)} \]

3. **Filling the Hundreds Place**:
The hundreds place \( A \) can be filled with any digit from 1 to 9 (since it cannot be 0 for a 3-digit number). Therefore, there are 9 possible choices for \( A \):
\[ A \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \quad \text{(9 ways)} \]

4. **Filling the Tens Place**:
The tens place \( B \) can be filled with any digit from 0 to 9. Since repetition is allowed, there are 10 possible choices for \( B \):
\[ B \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \quad \text{(10 ways)} \]

5. **Calculating the Total Number of Combinations**:
To find the total number of 3-digit numbers, we multiply the number of choices for each digit:
\[ \text{Total Ways} = \text{(Ways for A)} \times \text{(Ways for B)} \times \text{(Ways for C)} \]
Substituting the values we found:
\[ \text{Total Ways} = 9 \times 10 \times 1 = 90 \]

### Final Answer:
The total number of 3-digit numbers that can be formed with the unit digit as zero and repetition of digits allowed is **90**.