Integers Containing a Zero

Count the complement. A $k$-digit positive integer with no digit $0$ has each of its $k$ digits chosen from $\{1,\dots,9\}$. Every such string is a genuine $k$-digit number, because its leading digit is not zero. So there are $9^k$ of them.

The integers from $1$ to $9999$ are exactly the integers with one to four digits. The number $10000$ contains a zero, so it is not zero-free. The number of zero-free integers in the range is therefore

Subtracting this from the $10000$ integers in the range gives

Key idea. “At least one $0$“ is awkward to count directly. ”No $0$” is easy if the numbers are split by length, because then the leading-digit restriction takes care of itself.