[madnak]

Oh, another way to prove it is to look at how many repeats there are for each digit.

So for 1, there's 011, 111, 211, 311... and there's 110, 111, 112, 113... and there's 101, 111, 121, 131... There are ten numbers in each group. 30 numbers altogether. But 111 appears in all three groups - we don't want to count it three times, only once, so we really have 28 unique numbers.

Since there are only 3 digits in each number, only one of the digits can repeat. Therefore, every digit will repeat the same number of times through the whole 1000 possible 3-digit numbers. We know this number is 28, and there are 10 digits, so the total number of repeats is 280. There are 1000 numbers altogether, so there are 1000-280=720 numbers that don't repeat.