How many numbers with different digit each greater than 4000 can be formed from the digits 0,2,5,7,8 ?

Options

A. 160
B. 168
C. 320
D. 270

Answer

To solve the problem of how many numbers with different digits greater than 4000 can be formed from the digits 0, 2, 5, 7, and 8, we can break it down into two parts: counting the 4-digit numbers and counting the 5-digit numbers.

### Step 1: Count the 4-digit numbers greater than 4000

1. **Determine the first digit**:
- The first digit must be greater than 4 to ensure the number is greater than 4000. The possible digits for the first position are 5, 7, and 8.
- This gives us **3 choices** for the first digit.

2. **Choose the remaining digits**:
- After selecting the first digit, we have 4 digits left (including 0) to fill the remaining three positions.
- For the second position, we can choose any of the remaining 4 digits.
- For the third position, we can choose from the remaining 3 digits.
- For the fourth position, we can choose from the remaining 2 digits.

3. **Calculate the total combinations for 4-digit numbers**:
- The total number of combinations can be calculated as:
\[ \text{Total 4-digit numbers} = 3 \times 4 \times 3 \times 2 = 72 \]

### Step 2: Count the 5-digit numbers greater than 4000

1. **Determine the first digit**:
- For a 5-digit number, the first digit can be 2, 5, 7, or 8 (0 cannot be the first digit).
- This gives us **4 choices** for the first digit.

2. **Choose the remaining digits**:
- After selecting the first digit, we have 4 remaining digits to fill the other four positions.
- For the second position, we can choose from the remaining 4 digits.
- For the third position, we can choose from the remaining 3 digits.
- For the fourth position, we can choose from the remaining 2 digits.
- For the fifth position, we can choose from the remaining 1 digit.

3. **Calculate the total combinations for 5-digit numbers**:
- The total number of combinations can be calculated as:
\[ \text{Total 5-digit numbers} = 4 \times 4 \times 3 \times 2 \times 1 = 96 \]

### Step 3: Combine the results

- Finally, we add the total number of 4-digit numbers and 5-digit numbers:
\[ \text{Total numbers greater than 4000} = 72 + 96 = 168 \]

Thus, the total number of different digit numbers greater than 4000 that can be formed from the digits 0, 2, 5, 7, and 8 is **168**.