How many total increasing numbers are there? I've been asked this question and I'm close but just can't figure it out.
To clarify, an "increasing number" has to have increasing digits from left to right and cannot repeat digits anywhere else in the number.  
So an increasing number would be considered 12, or 125, or 2469; however, numbers that repeat certain digits such as 112 and/or numbers that fall such as 453 or 5498 are not increasing numbers.
An increasing number cannot begin with 0.
Single digits do count as an increasing number (1 or 9 would both be considered increasing numbers).
With the largest increasing number being 123456789, how many total increasing numbers are there?

The trivial 1 digit numbers... there are 9 of them.
There are ${9\choose 2}, 2$ digit numbers.
${9\choose n}, n$ digit numbers with $1\le n\le 9$
$\sum_\limits{n=0}^9 {9\choose n} = 2^9$
Since $0$ is not a possibility.
$\sum_\limits{n=1}^9 {9\choose n} = 2^9 - 1$
