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 in $2024$ is displayed as $20240426$. For how many dates in $2024$ will each digit appear an even number of times in the $8$-digit display for that date, with the exception that digit $2$ must appear an odd number of times?