Puzzle #117: Luggage Combinations

Is your luggage combination 1234? There are jokes about that.

It would be a bad idea to select too obvious a combination, whatever that amounts to. We could make up all sorts of rules about what combinations are obvious. Limiting it to the following, how many supposedly unobvious combinations are left? It is too obvious if the combination:

1) has all the same digit or three of the same digit,
2) has two pairs of digits regardless of the order,
3) has the digits in ascending order or descending order, or
4) forms an obvious arithmetic equation. Examples of these are: 7815 (7 + 8 = 15), 1798 (17 – 9 = 8), 9654 (9 × 6 = 54), and lastly 8199 (81 ÷ 9 = 9). Consider leading zeroes so 2024 is out (2 + 02 = 4).

There are unobvious combinations, but how many?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, September 2, 2015 at noon Pacific Time. I will post the answer shortly after.