To solve the problem of finding the number of positive integers that can be formed using the digits 0, 1, 2, 3, 4, and 5 without repeating any digit, and specifically how many of these integers are greater than 3000, we can break down the solution step by step.
### Step 1: Identify the Digits
The available digits are 0, 1, 2, 3, 4, and 5. Out of these, the digits 1, 2, 3, 4, and 5 are positive integers, while 0 cannot be the leading digit in any positive integer.
### Step 2: Count the Total Numbers Formed
We can form numbers of different lengths (1-digit, 2-digit, 3-digit, 4-digit, 5-digit, and 6-digit). We will calculate the total numbers for each case.
#### 1-Digit Numbers:
- The valid digits are 1, 2, 3, 4, 5.
- Total = 5 numbers.
#### 2-Digit Numbers:
- The first digit can be 1, 2, 3, 4, or 5 (5 choices).
- The second digit can be any of the remaining 5 digits (including 0).
- Total = 5 × 5 = 25 numbers.
#### 3-Digit Numbers:
- The first digit can be 1, 2, 3, 4, or 5 (5 choices).
- The second digit can be any of the remaining 5 digits.
- The third digit can be any of the remaining 4 digits.
- Total = 5 × 5 × 4 = 100 numbers.
#### 4-Digit Numbers:
- The first digit can be 1, 2, 3, 4, or 5 (5 choices).
- The second digit can be any of the remaining 5 digits.
- The third digit can be any of the remaining 4 digits.
- The fourth digit can be any of the remaining 3 digits.
- Total = 5 × 5 × 4 × 3 = 300 numbers.
#### 5-Digit Numbers:
- The first digit can be 1, 2, 3, 4, or 5 (5 choices).
- The second digit can be any of the remaining 5 digits.
- The third digit can be any of the remaining 4 digits.
- The fourth digit can be any of the remaining 3 digits.
- The fifth digit can be any of the remaining 2 digits.
- Total = 5 × 5 × 4 × 3 × 2 = 600 numbers.
#### 6-Digit Numbers:
- The first digit can be 1, 2, 3, 4, or 5 (5 choices).
- The second digit can be any of the remaining 5 digits.
- The third digit can be any of the remaining 4 digits.
- The fourth digit can be any of the remaining 3 digits.
- The fifth digit can be any of the remaining 2 digits.
- The sixth digit must be the last remaining digit.
- Total = 5 × 5 × 4 × 3 × 2 × 1 = 720 numbers.
### Step 3: Calculate the Total Numbers
Now, we add all the numbers formed:
Total = 5 (1-digit) + 25 (2-digit) + 100 (3-digit) + 300 (4-digit) + 600 (5-digit) + 720 (6-digit) = 1750.
### Step 4: Count Numbers Greater than 3000
To find how many of these numbers are greater than 3000, we can consider:
- **5-digit numbers**: All 600 are greater than 3000.
- **6-digit numbers**: All 720 are greater than 3000.
- **4-digit numbers**: The first digit must be 3, 4, or 5 to be greater than 3000.
#### 4-Digit Numbers Greater than 3000:
- If the first digit is 3: Remaining digits can be 0, 1, 2, 4, 5 (5 choices for the second digit, 4 for the third, and 3 for the fourth).
- If the first digit is 4: Remaining digits can be 0, 1, 2, 3, 5 (same calculation).
- If the first digit is 5: Remaining digits can be 0, 1, 2, 3, 4 (same calculation).
Total for 4-digit numbers:
- First digit = 3: 5 × 4 × 3 = 60
- First digit = 4: 5 × 4 × 3 = 60
- First digit = 5: 5 × 4 × 3 = 60
Total = 60 + 60 + 60 = 180.
### Final Count of Numbers Greater than 3000
Total numbers greater than 3000 = 600 (5-digit) + 720 (6-digit) + 180 (4-digit) = 1500.
### Conclusion
The total number of positive integers that can be formed using the digits 0, 1, 2, 3, 4, and 5, without repeating any digit, that are greater than 3000 is **1500**.
