How many numbers don’t contain the digit 9?

Let’s see how many numbers from $0$ to $10^n-1$ don’t contain the digit $9$.

For this kind of problem a computer scientist will most probably try to implement a sophisticated algorithm, a mathematician will try to find a direct formula, and a physicist will give an approximated value.

I’ll choose the math approach and instead of counting we can think to numbers as an array of digits. Imagine $n$ positions that can be filled with digits from $0$ to $9$. The digits with value $0$ from the front can be ignored. For example $00127$ represent actually then number $127$.

Considering the above each of the $n$ positions can be filled with a value from $0$ to $8$ because we don’t want the digit $9$. This means that we have $9$ ways to fill each position, resulting a total of $9^n$ numbers having up to $n$ digits and not containing the digit $9$. We also counted the number $0$ because it appears when all the $n$ positions are filled with digit $0$.