Three digit numbers are formed using the digits 0, 2, 4, 6, 8. A number is chosen out of these numbers at random. The probability that this number has same digits is

Options

A. `1/50`
B. `3/100`
C. `1/25`
D. `1/20`

Answer

To solve the problem, we need to find the probability that a randomly chosen three-digit number formed using the digits 0, 2, 4, 6, and 8 has all the same digits.

### Step-by-step Solution:

1. **Identify the digits available**:
The digits we can use to form the three-digit numbers are 0, 2, 4, 6, and 8.

2. **Determine the total possible outcomes**:
- A three-digit number cannot start with 0 (as it would then be a two-digit number).
- The first digit can be one of 2, 4, 6, or 8 (4 options).
- The second and third digits can be any of the 5 digits (0, 2, 4, 6, 8) since repetition is allowed.
- Therefore, the total possible outcomes can be calculated as follows:
\[ \text{Total Outcomes} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) = 4 \times 5 \times 5 = 100 \]

3. **Identify favorable outcomes**:
- We need to find the numbers where all three digits are the same.
- The possible three-digit numbers with the same digits can only be formed using the digits 2, 4, 6, and 8 (since 0 cannot be used).
- The valid numbers are: 222, 444, 666, and 888.
- Thus, there are 4 favorable outcomes.

4. **Calculate the probability**:
- The probability \( P \) that a randomly chosen three-digit number has all the same digits is given by the ratio of favorable outcomes to total possible outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{4}{100} = \frac{1}{25} \]

5. **Final Answer**:
The probability that the number has the same digits is \( \frac{1}{25} \).