Counting the number of strings that start and end with the same letter, are palindromes, or contain vowels only I have a question about this problem that I am trying to figure out:
A string of letters is called a palindrome, if reading the string from left to
right gives the same result as reading the string from right to left. For example, madam and racecar are palindromes. Recall that there are five vowels in the English alphabet: a, e, i, o, and u.

In this question, we consider strings consisting of 28 characters, with each character being a lowercase letter. Determine the number of such strings that 
  (i) start and end with the same letter, or 
  (ii) are palindromes, or 
  (iii) contain vowels only.

My answer so far:
$A = 26^{27}$,
$B = 26^{14}$,
$C = 5^{28}$
My question is, would I have to use a sum rule here or inclusion-exclusion rule? I am not sure about which method to use here. Any help would be appreciated. 

So, if I read the section that was boldface, now just a roman-numbered list, as meaning that if any of the given conditions are satisfied, then a string qualifies for the count, then you do indeed need to eliminate some double-counting, because these categories overlap.
However note that palindromes are a subset of words that start and end with the same letter, so in principle we don't need to count those separately at all; (ii) is completely contained within (i). So the only overlap that needs accounting for is: (iv) words consisting  of vowels only that start and end with the same letter (which will be result D in your scheme). 
Then your overall total is A + C - D.
