How many different 5 characters words are there with only one letter a?

I just need to clarify my answer to this exercise. This is a permutations exercise.

If we define a word to be a string of 5 letters of the English alphabet, regardless of meaning, then mnnnw is a word. How many different words are there?

In my answer I stated that since there is only one letter a per word then the remaining four letters will be one of the remaining letters from b - z. But the letter a can be in one of five places: first character, second character, third character, and so forth, leaving 25 possible characters (one of the letters from b - z) in the remaining four slots. So I would have $25 \cdot 25 \cdot 25 \cdot 25$ possible letters for the remaining letters (one per character slot). And since the letter a can be in one of 5 places I would have $25 \cdot 25 \cdot 25 \cdot 25 \cdot 5$ total words. But the study guide I am using says that there are $25 \cdot 25 \cdot 25 \cdot 25 \cdot 4$ total words. Is the book correct? If so, could you explain why?

Thanks,

Tony

Choosing one letter out of five to be a and the remaining 4 to be one of 25 choices can be done in $5 \cdot 25^4$ ways, just as you said.