If without repetition of the numbers, four-dight numbers are formed with the numbers 0, 2, 3 and 5, then the probability of such a number divisible by 5 is

Options

A. `(1)/(5)`
B. `(4)/(5)`
C. `(1)/(30)`
D. `(5)/(9)`

Explanation

To solve the problem of finding the probability of forming a four-digit number using the digits 0, 2, 3, and 5 that is divisible by 5, we will follow these steps:

### Step 1: Identify the total number of four-digit numbers that can be formed.

Since we are forming a four-digit number, the first digit cannot be 0 (as it would not be a four-digit number). Therefore, the first digit can only be 2, 3, or 5.

- **Choices for the first digit:** 3 options (2, 3, or 5)
- **Choices for the second digit:** 3 options (the remaining digits including 0)
- **Choices for the third digit:** 2 options (the remaining digits after choosing the first and second)
- **Choices for the fourth digit:** 1 option (the last remaining digit)

Calculating the total number of four-digit numbers:
\[ \text{Total numbers} = 3 \times 3 \times 2 \times 1 = 18 \]

### Step 2: Determine how many of these numbers are divisible by 5.

A number is divisible by 5 if its last digit is either 0 or 5. We will consider two cases based on the last digit.

#### Case 1: Last digit is 0
- **Choices for the first digit:** 2 options (2 or 3 or 5)
- **Choices for the second digit:** 2 options (the remaining digits excluding the first digit)
- **Choices for the third digit:** 1 option (the last remaining digit)

Calculating the total for this case:
\[ \text{Total for last digit 0} = 2 \times 2 \times 1 = 4 \]

#### Case 2: Last digit is 5
- **Choices for the first digit:** 2 options (2 or 3)
- **Choices for the second digit:** 2 options (the remaining digits including 0)
- **Choices for the third digit:** 1 option (the last remaining digit)

Calculating the total for this case:
\[ \text{Total for last digit 5} = 2 \times 2 \times 1 = 4 \]

### Step 3: Calculate the total number of four-digit numbers divisible by 5.

Adding the totals from both cases:
\[ \text{Total divisible by 5} = 4 + 4 = 8 \]

### Step 4: Calculate the probability.

The probability \( P \) of forming a four-digit number that is divisible by 5 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{8}{18} = \frac{4}{9} \]

### Final Answer:
The probability of forming a four-digit number that is divisible by 5 is \( \frac{4}{9} \).

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