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, New Year's Day in $2020$ is displayed as $20200101$. For how many dates in $2020$ 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?