To find the number of positive integers that can be formed using the given digits, we need to consider the different combinations and permutations of these digits. Let's break down the solution into steps:

Step 1: Count the number of ways to choose the digits.
Since we can use any number of digits from 0, 1, 2, 3, 4, 5, we need to count the number of ways to choose these digits. We can choose any number of digits from 0 to 5, so there are 2^6 = 64 possible choices. However, we need to subtract one choice where no digit is selected. Therefore, the number of ways to choose the digits is 64 - 1 = 63.

Step 2: Count the number of ways to arrange the chosen digits.
Once we have selected the digits, we need to count the number of ways to arrange them to form a positive integer. Since each digit can only be used once in each number, we have 6 choices for the first digit, 5 choices for the second digit, 4 choices for the third digit, and so on. Therefore, the number of ways to arrange the chosen digits is 6!.

Step 3: Multiply the results from Step 1 and Step 2.
To get the total number of positive integers, we multiply the number of ways to choose the digits (63) by the number of ways to arrange the chosen digits (6!). Therefore, the total number of positive integers is 63 * 6! = 63 * 720 = 45,360.

However, we need to consider that some of these numbers will have leading zeros, which are not considered as positive integers. To find the number of positive integers, we need to subtract the number of integers with leading zeros.

Step 4: Count the number of integers with leading zeros.
If a number starts with zero, it is not considered a positive integer. To find the number of integers with leading zeros, we fix the first digit as zero and count the number of ways to arrange the remaining digits. The remaining digits can be arranged in (6-1)! = 5! = 120 ways.

Step 5: Subtract the number of integers with leading zeros from the total number of integers.
To get the final answer, we subtract the number of integers with leading zeros (120) from the total number of integers (45,360). Therefore, the number of positive integers that can be formed is 45,360 - 120 = 45,240.

Therefore, the correct answer is option D) 45,240.