\section*{Problem 9 - V01209}

What percentage of
a) all 2-digit numbers
b) all 3-digit numbers
c) all 5-digit numbers
d) all 10-digit numbers
e) all 20-digit numbers
f) all 50-digit numbers

do not contain the digit 0?
}

For an \(n \geq 2\), there are exactly \(9 \cdot 10^{n-1} \mathrm{n}\)-digit numbers.

For an n-digit number to not contain a 0, each digit must be between 1 and 9, meaning there are \(9^{n}\) different n-digit numbers without a 0 in the digit sequence. The percentage share is thus

\[
\frac{9^{n}}{9 \cdot 10^{n-1}} \cdot 100 \% = 0.9^{n-1} \cdot 100 \%
\]

This results in: a) \(90 \%\), b) \(81 \%\), c) \(65.61 \%\), d) \(38.7 \%\), e) \(13.5 \%\), and f) \(0.57 \%\).