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\title{Question 4: Counting Five-Digit Numbers}
\author{}
\date{}
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\section*{1. How many five-digit numbers can be written?}
Let the number of digits be:
\[
K = 5.
\]
The set of all possible numbers in one digit is:
\[
D = \{0,1,\dots,9\}.
\]
The number of elements in \( D \) is:
\[
d = \#D = 10.
\]
Thus, the total number of five-digit numbers is:
\[
n = d^K = 100\,000.
\]
\subsection*{Other cases:}
\begin{itemize}
\item Excluding numbers with the most significant digit 0:
\[
n_{\,\backslash\,\{0\}} = (d-1)d^{K-1} = 90\,000.
\]
\item Including negative numbers:
\[
n_{\mathbb{Z}} = 2d^K - 1 = 199\,999.
\]
\item Including negative numbers but excluding numbers with the most significant digit 0:
\[
n_{\mathbb{Z}\,\backslash\,\{0\}} = 2(d-1)d^{K-1} = 180\,000.
\]
\end{itemize}
\textbf{Answer:}
\[
\begin{aligned}
n &= 100\,000, \\
n_{\,\backslash\,\{0\}} &= 90\,000, \\
n_{\mathbb{Z}} &= 199\,999, \\
n_{\mathbb{Z}\,\backslash\,\{0\}} &= 180\,000.
\end{aligned}
\]
\bigskip
\section*{2. How many five-digit numbers contain at least one even digit?}
Define the set of odd digits:
\[
D_{\text{odd}} = \{1,3,5,7,9\}.
\]
The number of elements in \( D_{\text{odd}} \) is:
\[
d_{\text{odd}} = 5.
\]
The number of five-digit numbers with only odd digits:
\[
n_{\text{odd}} = d_{\text{odd}}^K = 3\,125.
\]
Thus, the number of five-digit numbers containing at least one even digit:
\[
n_{\#\{\text{even} \geq 1\}} = n - n_{\text{odd}} = 96\,875.
\]
\subsection*{Other cases:}
\begin{itemize}
\item Excluding numbers with the most significant digit 0:
\[
n_{\,\backslash\,\{0\}_{\#\{\text{even} \geq 1\}}} = n_{\,\backslash\,\{0\}} - n_{\text{odd}} = 86\,875.
\]
\item Including negative numbers:
\[
n_{\mathbb{Z}_{\#\{\text{even} \geq 1\}}} = n_{\mathbb{Z}} - n_{\text{odd}} = 196\,874.
\]
\item Including negative numbers but excluding numbers with the most significant digit 0:
\[
n_{\mathbb{Z}\,\backslash\,\{0\}_{\#\{\text{even} \geq 1\}}} = n_{\mathbb{Z}\,\backslash\,\{0\}} - n_{\text{odd}} = 176\,875.
\]
\end{itemize}
\textbf{Answer:}
\[
\begin{aligned}
n_{\#\{\text{even} \geq 1\}} &= 96\,875, \\
n_{\,\backslash\,\{0\}_{\#\{\text{even} \geq 1\}}} &= 86\,875, \\
n_{\mathbb{Z}_{\#\{\text{even} \geq 1\}}} &= 196\,874, \\
n_{\mathbb{Z}\,\backslash\,\{0\}_{\#\{\text{even} \geq 1\}}} &= 176\,875.
\end{aligned}
\]
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