I think the answer is 378, based on the following logic:
- All single digit numbers are distinct in nature, so 9.
- All double digit numbers have distinct digits except 11,22,33,etc. So the total two digit numbers which have distinct digits are: 81 (99-10+1= 90 is the total number of two digit numbers and then you have 9 with repeated digits (11,22,33,...), so a total of 81 digits).
- Then we have three digit numbers, this deserves a detailed analysis:
Here, we use the product rule first to check the digits between 100-499:
The number of digits that can occupy the first place is 4 (1,2,3,4)
The number of digits that occupy the second place is 9 (All digits except for the one used above).
The number of digits that occupy the third place is 8 ( All digits except for the first two digits).
So the total numbers with distinct digits from 100 to 499 is 4 X 9 X 8 = 288.
- 500 itself fails to satisfy the claim, the reason being the repitition of the zeros.
Therefore, the answer is 288+9+81= 378.