The number of 4 digit numbers that can be formed using digits `0,1,2,3,4,5` (repetition allowed) which are greater than `4321` is

Options

A. 306
B. 310
C. 288
D. 280

Answer

To find the number of 4-digit numbers that can be formed using the digits `0, 1, 2, 3, 4, 5` (with repetition allowed) that are greater than `4321`, we can break down the problem step by step.

### Step 1: Analyze the first digit
The first digit of the number must be greater than `4` (the first digit of `4321`). The possible digits greater than `4` from the set `{0, 1, 2, 3, 4, 5}` are `5`.

- **Choices for the first digit:** 1 (only `5`)

### Step 2: Analyze the second digit
If the first digit is `5`, the second digit can be any digit from the set `{0, 1, 2, 3, 4, 5}`.

- **Choices for the second digit:** 6 (any digit)

### Step 3: Analyze the third digit
The third digit can also be any digit from the set `{0, 1, 2, 3, 4, 5}`.

- **Choices for the third digit:** 6 (any digit)

### Step 4: Analyze the fourth digit
The fourth digit can also be any digit from the set `{0, 1, 2, 3, 4, 5}`.

- **Choices for the fourth digit:** 6 (any digit)

### Step 5: Calculate total combinations for the first case
Now, we can calculate the total number of combinations when the first digit is `5`:

\[ \text{Total combinations} = 1 \times 6 \times 6 \times 6 = 216 \]

### Step 6: Analyze numbers starting with `4`
Next, we need to consider numbers that start with `4`. In this case, the second digit must be greater than `3` (the second digit of `4321`).

- Possible digits for the second digit greater than `3`: `4` and `5` (2 choices).

#### Case 1: Second digit is `4`
If the second digit is `4`, the third digit must be greater than `2` (the third digit of `4321`).

- Possible digits for the third digit greater than `2`: `3`, `4`, and `5` (3 choices).

For the fourth digit, it can be any digit from the set `{0, 1, 2, 3, 4, 5}` (6 choices).

So, the total combinations for this case are:

\[ \text{Total combinations} = 1 \times 1 \times 3 \times 6 = 18 \]

#### Case 2: Second digit is `5`
If the second digit is `5`, the third digit can be any digit from the set `{0, 1, 2, 3, 4, 5}` (6 choices).

- **Choices for the third digit:** 6 (any digit)

For the fourth digit, it can also be any digit from the set `{0, 1, 2, 3, 4, 5}` (6 choices).

So, the total combinations for this case are:

\[ \text{Total combinations} = 1 \times 1 \times 6 \times 6 = 36 \]

### Step 7: Combine all cases
Now, we can combine all the cases:

1. Numbers starting with `5`: 216
2. Numbers starting with `4` and second digit `4`: 18
3. Numbers starting with `4` and second digit `5`: 36

Total numbers greater than `4321`:

\[ \text{Total} = 216 + 18 + 36 = 270 \]

### Final Answer
The total number of 4-digit numbers that can be formed using the digits `0, 1, 2, 3, 4, 5` (with repetition allowed) that are greater than `4321` is **270**.