Three digital numbers are formed using the digits 0.2.4.6.8. A number is chosen at random out of these numbers. What is the probability that the number has the same digit?

Options

A. `1//16`
B. `1//25`
C. `16//25`
D. `1//645`

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 1: Determine the total number of three-digit numbers that can be formed.
- The first digit (hundreds place) cannot be 0 (to ensure it's a three-digit number). Therefore, the possible digits for the first digit are 2, 4, 6, or 8. This gives us 4 options.
- The second digit (tens place) can be any of the digits: 0, 2, 4, 6, or 8. This gives us 5 options.
- The third digit (units place) can also be any of the digits: 0, 2, 4, 6, or 8. This gives us 5 options.

Now, we can calculate the total number of three-digit numbers:
\[ \text{Total numbers} = (\text{choices for first digit}) \times (\text{choices for second digit}) \times (\text{choices for third digit}) = 4 \times 5 \times 5 = 100. \]

### Step 2: Determine the number of favorable outcomes (numbers with the same digit).
- The only digits that can form a three-digit number with all the same digits from our set are 2, 4, 6, and 8.
- Therefore, the favorable outcomes are: 222, 444, 666, and 888. This gives us 4 favorable outcomes.

### Step 3: Calculate the probability.
- The probability \( P \) is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{100} = \frac{1}{25}. \]

### Final Answer:
The probability that a randomly chosen three-digit number has all the same digits is \( \frac{1}{25} \).

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