How many different 4 - digit numbers are there which have the digits 1,2,3,6,7 and 0 such that always the digit 3 appears exactly once in the number ? (repetition is allowed )

Options

A. 350
B. 300
C. 500
D. 425

Answer

To solve the problem of finding how many different 4-digit numbers can be formed using the digits 1, 2, 3, 6, 7, and 0, with the condition that the digit 3 appears exactly once, we can break it down into several cases based on the position of the digit 3.

### Step-by-Step Solution:

1. **Identify the Digits**:
The available digits are 1, 2, 3, 6, 7, and 0. Since we are forming a 4-digit number, the first digit cannot be 0.

2. **Case 1: Digit 3 is in the 1st Position**:
- The number looks like: **3 _ _ _**
- The remaining three positions can be filled with any of the digits (1, 2, 6, 7, 0) since repetition is allowed.
- Choices for the 2nd, 3rd, and 4th positions: 5 choices each.
- Total combinations for this case:
\[ 5 \times 5 \times 5 = 125 \]

3. **Case 2: Digit 3 is in the 2nd Position**:
- The number looks like: **_ 3 _ _**
- The first digit can be any of (1, 2, 6, 7) (0 cannot be here).
- Choices for the 1st position: 4 choices (1, 2, 6, 7).
- Choices for the 3rd and 4th positions: 5 choices each.
- Total combinations for this case:
\[ 4 \times 5 \times 5 = 100 \]

4. **Case 3: Digit 3 is in the 3rd Position**:
- The number looks like: **_ _ 3 _**
- The first digit can again be any of (1, 2, 6, 7).
- Choices for the 1st position: 4 choices (1, 2, 6, 7).
- Choices for the 2nd and 4th positions: 5 choices each.
- Total combinations for this case:
\[ 4 \times 5 \times 5 = 100 \]

5. **Case 4: Digit 3 is in the 4th Position**:
- The number looks like: **_ _ _ 3**
- The first digit can be any of (1, 2, 6, 7).
- Choices for the 1st position: 4 choices (1, 2, 6, 7).
- Choices for the 2nd and 3rd positions: 5 choices each.
- Total combinations for this case:
\[ 4 \times 5 \times 5 = 100 \]

6. **Total Combinations**:
Now, we sum the combinations from all four cases:
\[ 125 + 100 + 100 + 100 = 425 \]

### Final Answer:
The total number of different 4-digit numbers where the digit 3 appears exactly once is **425**.

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