Sequence of letters of the form abcba A sequence of letters of the form abcba in which the sequence is unchanged upon reversing
the order is called a palindrome. How many 5 letter palindromes are there if we are allowed to repeat letters more than twice?
When i tried to do this problem out i thought would use the multiplication rule and say 26 factorial but i am having a hard time thinking of a way to meet all of the qualifications of the problem. 
 A: Hint:  if you choose the first letter, you know the last one.  How many letters can you choose?
A: I believe it would just be 26*26*26. You have 26 letters for the 'a', 26 for 'b', and 26 for 'c'. Because you can repeat letters more than twice, a factorial wouldn't work, because you'd be removing a letter you could use for both of the other spots. 
So, 26*26*26=17576.
A: We are interested in 5-letter palindromes.  Presumably the alphabet we use will be the standard $26$-letter alphabet: "A,B,C,D,...,Z"
We have that they will be of the form:
$$\underline{\color{red}{\bullet}}~\underline{\color{blue}{\bullet}}~\underline{\color{green}{\bullet}}~\underline{\color{blue}{\bullet}}~\underline{\color{red}{\bullet}}$$
Since it is a palindrome, we know that the first character must be the same as the last character.  Similarly the second character must be the same as the fourth character.  (as suggested by my coloring scheme above)
Break this up via multiplication principle.


*

*How many ways can you choose the letter used for the red location above?

*How many ways can you choose the letter used for the blue location above?

*How many ways can you choose the letter used for the green location above?
A: The answer is quite obvious. As you can repeat a letter more than twice, the answer is $26^3$. In general, for $n$ letter palindromes, the expression would be $26^{\lfloor \frac{n+1}{2} \rfloor}$.
