A [palindrome](https://artofproblemsolving.com/wiki/index.php/Palindrome) is a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) that reads the same forwards and backwards.  If one neglects the colon, certain times displayed on a digital watch are palindromes.  Three examples are: $\boxed{1:01}$, $\boxed{4:44}$, and $\boxed{12:21}$.  How many times during a $12$-hour period will be palindromes?
$\text{(A)}\ 57 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 63 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 93$
From $1$ to $9$, the times will be of the form $\boxed{a:ba}$.  There are $9$ choices for $a$ and $6$ choices for $b$, so there are $9\cdot 6 =54$ times in this period.
From $10$ to $12$, the minutes are already determined, so there are only $3$ times in this case.
In total, there are $54+3=57\rightarrow \boxed{\text{A}}$ palindromic times.