The number of all positive integers up to 500 with non-repeating digits is:

The correct answer is 378.

To find the number of all positive integers up to 500 with non-repeating digits, we can categorize the numbers by the number of digits they contain: one-digit numbers, two-digit numbers, and three-digit numbers up to 500.

1. One-Digit Numbers (1 to 9):
All one-digit positive integers naturally have non-repeating digits.
These numbers are 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Total number of one-digit integers = 9.

2. Two-Digit Numbers (10 to 99):
Let a two-digit number be represented in the form XY, where X is the tens digit and Y is the units digit.
For the digits to be non-repeating:
- The tens digit X can be any non-zero digit from 1 to 9. Thus, there are 9 choices for X.
- The units digit Y can be any digit from 0 to 9, except it cannot be equal to X. Thus, there are 9 remaining choices for Y.
Using the fundamental counting principle, the total number of two-digit integers with non-repeating digits is:

$9\times 9=81$

3. Three-Digit Numbers (100 to 500):
Since the limit is 500, we can divide this count into two parts: numbers from 100 to 499, and the number 500 itself.

Case A: Numbers from 100 to 499:
Let a three-digit number be represented in the form XYZ, where X is the hundreds digit, Y is the tens digit, and Z is the units digit.
- The hundreds digit X can only be 1, 2, 3, or 4 (since the number must be less than 500). This gives 4 choices.
- The tens digit Y can be any digit from 0 to 9, excluding the digit chosen for X. This gives 9 choices.
- The units digit Z can be any digit from 0 to 9, excluding the digits chosen for X and Y. This leaves 8 choices.
The total number of such three-digit integers is:

$4\times 9\times 8=288$

Case B: The number 500:
The number 500 has repeating digits (the digit 0 is repeated). Therefore, it is not included in our count, contributing 0 to the total.

Total Count:
Adding the results from all the cases together, we get the total number of positive integers up to 500 with non-repeating digits:

$\text{Total}=9+81+288=378$