2009 AMC 10B Problem 11

A $7$-digit palindrome has the form with the middle digit used once and the outer three digits each used twice. Only $5$ appears an odd number of times, so $5$ must be the middle digit. The remaining digits $2,3,5$ fill the first three positions in some order and mirror to the last three. There are $3!=6$ such orderings. Thus, the correct answer is A.