Ascending Decimal Digit Puzzle

[K Sengupta]

A positive decimal (base 10) integer G is called an ascending number if the number of digits in G is at least 2, all the digits of G are strictly ascending, and G does not contain any leading zeroes.

For example, the number 1236789 is an ascending number. However, the number 1222333555666 is not an ascending number, since the digits corresponding to the said number are not strictly ascending. Similarly, 00123456 is not an ascending number, since it contains leading zeroes.

Determine the total number of ascending numbers in the decimal (base 10) system.

[humanino]

Consider a stairway with 10 steps. Assume you can skip as many steps as you want. One choice of step combination to climb this step can be labeled by a unique ascending number. Reciprocally, any ascending number can be uniquely associated to such a choice.

I thought the stairway riddle has already been posted. Otherwise, you can solve it with an arbitrary number of steps. The answer equals the number of ascending numbers in the corresponding base :smile:

[jimmysnyder]

Let N be the number of ascending numbers using the digits 1 through 8. Tack a 9 onto the end of each one of them. They are also ascending numbers and are all of them except for 19, 29, ..., 89. So the answer is 2N + 8.
Let M be the number of ascending numbers using the digits 1 through 7. Tack an 8 onto the end of each one of them.
Etc.

[daskalou]

Is it 502?

The formula for this would be:

2^n - n - 1

where n = the number of USABLE digits, which in this case is 9 (since the digit 0 is useless in this situation)