AMC 12 2023a Problem 7

A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as $20230428$. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?

\textbf{(A)}~5 \textbf{(B)}~6 \textbf{(C)}~7 \textbf{(D)}~8 \textbf{(E)}~9