Q:

How can I count the number of $n$ digit positive integers without a specific digit?

Came across the Kempner Series and was doing a little reading.
The proof that the Kempner Series is bounded by 80 requires the fact that the number of $n$ digit positive integers without the digit 9 in them is $8(9^{n-1})$.
I bet there is a combinatorial way to show this, but I can't think of it.  I thought of proving it with induction, but that seems like overkill.
How can I prove this to myself?

A:

hint
In a $n$ digit number, you cannot use $9$ in all positions, and cannot use $0$ in the first position.

how many choices for first position?
how many choices for each of the other positions?
how many total choices?

