Four digit numbers are formed using the digits 0, 2, 3,5 without repetition. The probability of such a number divisible by 5 is

Options

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

Answer

To solve the problem of finding the probability that a four-digit number formed using the digits 0, 2, 3, and 5 (without repetition) is divisible by 5, we can follow these steps:

### Step 1: Determine the total number of four-digit numbers that can be formed.
- The first digit cannot be 0 (as it would not be a four-digit number). Therefore, we have three options for the first digit: 2, 3, or 5.
- After choosing the first digit, we have three digits left, and we can fill the remaining three positions with the remaining digits.

**Calculations:**
1. Choose the first digit: 3 options (2, 3, or 5).
2. Choose the second digit: 3 remaining options (including 0).
3. Choose the third digit: 2 remaining options.
4. Choose the fourth digit: 1 remaining option.

So, the total number of four-digit numbers is:
\[ 3 \times 3 \times 2 \times 1 = 18 \]

### Step 2: Determine the number of favorable outcomes (numbers divisible by 5).
A number is divisible by 5 if it ends with either 0 or 5. We will consider two cases:

#### Case 1: The number ends with 0.
- If the last digit is 0, the first digit can be either 2 or 3 or 5 (but not 0). So we have 3 options for the first digit.
- After choosing the first digit, we have 2 remaining digits to fill the second and third positions.

**Calculations:**
1. Choose the first digit: 3 options (2, 3, or 5).
2. Choose the second digit: 2 remaining options.
3. Choose the third digit: 1 remaining option.

So, the total number of four-digit numbers ending with 0 is:
\[ 3 \times 2 \times 1 = 6 \]

#### Case 2: The number ends with 5.
- If the last digit is 5, the first digit can only be 2 or 3 (not 0 or 5). So we have 2 options for the first digit.
- After choosing the first digit, we have 2 remaining digits (including 0) to fill the second and third positions.

**Calculations:**
1. Choose the first digit: 2 options (2 or 3).
2. Choose the second digit: 2 remaining options.
3. Choose the third digit: 1 remaining option.

So, the total number of four-digit numbers ending with 5 is:
\[ 2 \times 2 \times 1 = 4 \]

### Step 3: Calculate the total number of favorable outcomes.
Now, we add the favorable outcomes from both cases:
\[ 6 \text{ (ending with 0)} + 4 \text{ (ending with 5)} = 10 \]

### Step 4: Calculate the probability.
The probability \( P \) that a randomly formed four-digit number is divisible by 5 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{18} = \frac{5}{9} \]

### Final Answer:
The probability that a four-digit number formed using the digits 0, 2, 3, and 5 without repetition is divisible by 5 is \( \frac{5}{9} \).

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