Am I reading this combination question correctly? The question is:

If any ten letters (of the $27$-letter Spanish alphabet) could be used to form ten-letter words, how many words would have at least one repeated
  letter?

The answer I give is $10^{10}$ (given ten letters?), however, should it be $27^{10}$ (choosing each letter from the entire alphabet?)? 
 A: All "words" of ten letters would mean a free choice of any letter for each position:  
$27\times 27\times \cdots = 27^{10}$
Words with no repeats means the choice gradually diminishes, a selection without replacement, 
$27\times 26\times 25 \times\cdots\times 18= \dfrac{27!}{17!}$
Words with at least one repeat is those in the first group but not in the second,
$27^{10} - \dfrac{27!}{17!}$
A: Not sure what you are trying to count when you say $10^{10}$.
$27^{10}$ is the number of 10 letter words you can form with repetitions allowed.
The problem here you want at least one repeated letter. When you think of all possibilities, you realize you have to consider (a) one repetition (b) two repetitions (c) three repetitions ..... Also you'll have to consider multiple alphabets repeating multiple times. This is quite difficult to exhaustively enumerate. Is there an easier way?
There is. Number of ways with at least one repetition = number of total ways - number of ways with no repetition = $27^{10} - \binom{27}{10}\cdot 10!$. Much easier...
