How many numbers of 5 digits can be formed with the digits 0,2,3,4 and 5 if the digits may repeat ?

Options

A. a)2500
B. b)250
C. c)120
D. d)2400

Answer

To solve the problem of how many 5-digit numbers can be formed using the digits 0, 2, 3, 4, and 5 with repetition allowed, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Digits**: The available digits are 0, 2, 3, 4, and 5.

2. **Determine the Number of Places**: Since we want to form a 5-digit number, we will have 5 places to fill.

3. **Consider the First Digit**: The first digit of a 5-digit number cannot be 0 (as it would then not be a 5-digit number). Therefore, the valid options for the first digit are 2, 3, 4, and 5. This gives us 4 choices for the first digit.

4. **Fill the Remaining Digits**: The remaining 4 digits can be filled with any of the 5 available digits (including 0). Since repetition is allowed, each of these 4 places can be filled in 5 ways.

5. **Calculate the Total Combinations**:
- For the first digit: 4 choices (2, 3, 4, 5)
- For the second digit: 5 choices (0, 2, 3, 4, 5)
- For the third digit: 5 choices (0, 2, 3, 4, 5)
- For the fourth digit: 5 choices (0, 2, 3, 4, 5)
- For the fifth digit: 5 choices (0, 2, 3, 4, 5)

Therefore, the total number of 5-digit numbers can be calculated as:
\[ \text{Total} = 4 \times 5 \times 5 \times 5 \times 5 \]

6. **Perform the Calculation**:
\[ \text{Total} = 4 \times 5^4 = 4 \times 625 = 2500 \]

### Final Answer:
The total number of 5-digit numbers that can be formed is **2500**.

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